Quantum Pendulum via Physics-Informed Neural Networks: A Topological Approach to Quantum Dynamics

Pêndulo Quântico via Redes Neurais Informadas por Física: Uma Abordagem Topológica para

Abstract

The numerical simulation of quantum systems with non-trivial topology typically requires specialized coordinate charts or symplectic integrators. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative, but standard implementations struggle with periodic boundary conditions. We present a robust PINN architecture for the Quantum Pendulum (a particle on $S^1$) that enforces strict periodicity via a Trigonometric Coordinate Embedding layer. This “Hard Constraint” approach eliminates the need for unstable boundary penalty terms. We successfully reconstruct the unitary evolution of the Mathieu ground state under a non-linear cosine potential, achieving a norm conservation $>0.99$ and initial condition error of $𝒪\left({10}^{-6}\right)$. The method captures subtle quantum phenomena, including macroscopic tunneling tails and phase space delocalization. Validation is performed against the exact analytical solution via Mathieu functions. This work bridges the gap between modern Scientific Machine Learning and canonical quantum mechanics, serving as a tutorial on embedding topological constraints into neural solvers.

Keywords:
Physics-Informed Neural Networks; Quantum Pendulum; Topological Constraints; Schrödinger Equation; Scientific Machine Learning

Resumo

A simulação numérica de sistemas quânticos com topologia não trivial tipicamente requer cartas de coordenadas especializadas ou integradores simpléticos. Redes Neurais Informadas por Física (Physics-Informed Neural Networks - PINNs) oferecem uma alternativa livre de malhas, mas implementações padrão enfrentam dificuldades com condições de contorno periódicas. Apresentamos uma arquitetura robusta de PINN para o Pêndulo Quântico (uma partícula em $S^1$) que impõe periodicidade estrita via uma camada de Embedding de Coordenadas Trigonométricas. Esta abordagem de “Restrição Rígida” (Hard Constraint) elimina a necessidade de termos de penalidade de contorno instáveis. Reconstruímos com sucesso a evolução unitária do estado fundamental de Mathieu sob um potencial cosseno não linear, alcançando uma conservação de norma $>0.99$ e erro na condição inicial de $𝒪\left({10}^{-6}\right)$. O método captura fenômenos quânticos sutis, incluindo caudas de tunelamento macroscópico e deslocalização no espaço de fase. A validação é realizada contra a solução analítica via funções de Mathieu. Este trabalho preenche a lacuna entre o moderno Aprendizado de Máquina Científico e a mecânica quântica canônica, servindo como um tutorial sobre a incorporação de restrições topológicas em solvers neurais.

Palavras-chave:
Redes Neurais Informadas por Física; Pêndulo Quântico; Restrições Topológicas; Equação de Schrödinger; Aprendizado de Máquina em Física

1. Introduction

The computational resolution of the Schrödinger equation stands as a pillar of modern physics. Since the pioneering work of Lagaris et al. (1998) [¹], who first proposed utilizing neural networks to solve differential equations, the field has evolved drastically. Recently, Carleo and Troyer [²] initiated the “Quantum Machine Learning” revolution by introducing Neural Quantum States (NQS), demonstrating that neural networks can efficiently represent the exponentially complex wavefunctions of many-body systems.

However, a terminological dichotomy persists. The majority of Machine Learning (ML) applications in physics remain Data-Driven: they require vast datasets generated by classical solvers (such as Crank-Nicolson or Finite Elements) to train a surrogate model [³]. This reliance on external data is frequently expensive and epistemologically circular: one must solve the problem numerically to subsequently teach the network to mimic it.

For the reader originating from the physical sciences, it is useful to define that a neural network is, essentially, a universal function approximator comprised of layers of interconnected neurons. Each neuron processes inputs via linear transformations (weights and biases) followed by non-linear activation functions. Learning occurs via iterative optimization of these parameters to minimize a loss function, typically employing gradient-based algorithms such as backpropagation.

Within this landscape, the paradigm of Scientific Machine Learning (ScML) emerges, whose primary tool is the Physics-Informed Neural Network (PINN), formalized by Raissi et al. [⁴]. Unlike purely statistical models, PINNs are evaluated not merely against labeled data, but in strict compliance with the laws of physics. This is achieved by embedding the local residuals of the governing partial differential equation (PDE) directly as penalty terms within the global loss function. Consequently, the network learns the PDE solution by minimizing the structural error of the system’s own physics [⁵]. While NQS rely on Variational Monte Carlo focusing strictly on the many-body wavefunction ansatz [²], PINNs aim to approximate the continuous spatiotemporal solution of classical and quantum PDEs, intentionally avoiding the use of discrete grids or prior global label generation.

Nevertheless, PINN applications have dominantly succeeded in topologically trivial domains (such as the Harmonic Oscillator in $ℝ$[⁶]). Systems defined on compact manifolds, such as the Quantum Pendulum on the $S^1$ circle, pose a rigorous periodic constraint problem. The standard practice of adding “Soft Constraints”, i.e., boundary condition (BC) penalty terms appended to the loss function, frequently presents numerical instabilities, since the network may satisfy local physics without respecting the global topology of the unfolded space [⁷, ⁸].

In this work, we present a didactic tutorial on the implementation of PINNs for the Quantum Pendulum, confronting the topological challenge via the Periodic Embedding method proposed by Dong and Ni [⁸]. This “Hard Constraint” approach treats the coordinates not as points on a line bounded by artificial borders, but as an $S^1$ manifold immersed in ${ℝ}^2$. The overarching system serves as an ideal laboratory to explore Macroscopic Quantum Tunneling (MQT) [⁹] and unitary evolution under non-linear potentials. We aim to deliver accessible yet rigorous material for those entering topological ScML. To ensure reproducibility, the complete source code is openly available in a public repository: https://github.com/jjcarlosb-hue/quantum-pendulum-pinn.

The remainder of this article is organized as follows: Section 2 details the physical framework of the quantum pendulum, the quantization process on the $S^1$ circle, and the proposed topological Embedding architecture. Section 3 presents the training results, evaluating energy and norm conservation, as well as benchmarking the neural prediction against the analytical solution via Mathieu functions. Finally, Section 4 articulates the current advantages and limitations of the method concerning applications in quantum circuits and superconducting qubits. A comparative summary of these paradigms is presented in Table 1.

Table 1
Comparison of machine learning approaches for quantum wavefunctions. Our work focuses on the PINN class targeting closed spatial domains ($S^1$) with rigid topological enforcement via Embedding, eliminating iterative penalizations (Soft Constraints).

Work	System	Topology	Constraint Type	Learning Paradigm
Carleo & Troyer (2017) [2]	Spin Chains	Discrete	N/A	Variational Monte Carlo
Raissi et al. (2019) [4]	Schrödinger Equation	${ℝ}^n$ (Line)	Soft (Penalty)	Physics-Informed
Jin et al. (2020) [6]	Harmonic Osc.	$ℝ$ (Line)	Data Loss	Data-Driven (Noisy)
This Work	Quantum Pendulum	$S^1$ (Circle)	Hard (Embedding)	Physics-Informed

2. Methodology

2.1. Physical modeling: from the classical to the quantum pendulum

The planar pendulum is a prototypical system in both classical and quantum mechanics, representing a particle of mass $m$ constrained to move on a circle of radius $l$ in a uniform gravitational field $g$. In the classical regime, its instantaneous state is fully specified by the angular displacement $\theta$ and the angular velocity $\dot{\theta }$. Following standard Lagrangian formulation [¹⁰], the kinetic energy $T$ and the potential energy $V$ are:

(1) $$T=\frac{1}{\text{2}}m{l}^2{\dot{\theta }}^2,V\left(\theta \right)=mgl\left(1-\cos \theta \right).$$

The dynamics are governed by the Lagrangian $L=T-V$. The canonical momentum $p_{\theta }$, which in this rotational geometry corresponds to the orbital angular momentum $L_z$ along the axis of rotation, is defined as:

(2) $$p_{\theta }=\frac{\partial L}{\partial \dot{\theta }}=m{l}^2\dot{\theta }=I\dot{\theta },$$

where $I=m{l}^2$ is the pendulum’s moment of inertia. The classical Hamiltonian, $\mathscr{H}=p_{\theta }\dot{\theta }-L$, represents the total energy of the conservative system:

(3) $$\mathscr{H}=\frac{p_{\theta }^2}{2I}+mgl\left(1-\cos \theta \right).$$

2.1.1. Canonical quantization on the circle $S^1$

Transitioning to the quantum regime necessitates the canonical quantization procedure [¹¹]. The classical displacement $\theta$ and momentum $p_{\theta }$ are promoted to Hermitian operators $\widehat{\theta }$ and ${\widehat{p}}_{\theta }$ acting upon a Hilbert space ${ℒ}^2\left(S^1\right)$ of wavefunctions $\text{Ψ}(\theta ,t)$. Here, $\hslash \approx 1.054\times {10}^{-34}$ J$\cdot$s is the reduced Planck constant, which quantizes fundamental action. Owing to the periodic topology of the circle, the fundamental commutation relation requires additional care regarding the global angular momentum, being expressed via the single-valued operator $e^{i\widehat{\theta }}$[¹²]:

(4) $$[e^{i\widehat{\theta }},{\widehat{p}}_{\theta }]=-\hslash e^{i\widehat{\theta }},{\widehat{p}}_{\theta }=-i\hslash {\partial }_{\theta }.$$

In the position representation, the angular momentum operator is denoted as ${\widehat{p}}_{\theta }=-i\hslash \frac{\partial }{\partial \theta }$. A critical feature of the pendulum is its configuration space: the $S^1$ circle. For the wavefunction to be physically acceptable and single-valued under a full $2\pi$ rotation, it must satisfy the cyclic boundary condition:

(5) $$\text{Ψ}(\theta +2\pi ,t)=\text{Ψ}(\theta ,t).$$

As highlighted by Griffiths [¹³], this topological restriction leads to the quantization of the ${\widehat{p}}_{\theta }$ eigenvalues as $n\hslash$ ($n\in \mathbb{Z}$), mirroring the quantization of angular momentum in atomic systems. (Adopting the convention expressed by Carruthers and Nieto [¹²], the negative sign naturally arises from the action of ${\widehat{p}}_{\theta }=-i\hslash \partial /\partial \theta$ on the angular eigenstates.) The dynamics are subsequently governed by the Time-Dependent Schrödinger Equation (TDSE):

(6) $$i\hslash \frac{\partial \text{Ψ}}{\partial t}=\left[-\frac{{\hslash }^2}{2I}\frac{{\partial }^2}{{\partial \theta }^2}+mgl\left(1-\cos \theta \right)\right]\text{Ψ}(\theta ,t).$$

2.1.2. Nondimensional symmetrization andMathieu correspondence

For numerical implementation and better generalization across physical scales, we adopt a dimensionless form by scaling variables such that $\hslash =1$ and expressing energies in units of the inertial energy scale ${ℰ}_r={\hslash }^2/I$. Setting $I=1$, the TDSE simplifies to:

(7) $$i\frac{\partial \text{Ψ}}{\partial t}=-\frac{1}{\text{2}}\frac{{\partial }^2\text{Ψ}}{{\partial \theta }^2}+\kappa \left(1-\cos \theta \right)\text{Ψ}(\theta ,t),$$

where $\kappa =mgl/{ℰ}_r$ is the dimensionless potential strength.

2.1.3. Real-imaginary decomposition for neural implementation

Because standard neural network architectures naturally operate on real-valued fields, the complex wavefunction $\text{Ψ}(\theta ,t)=u(\theta ,t)+iv(\theta ,t)$ must be decomposed into its real and imaginary components. Operating on the dimensionless TDSE and separating the components yields a coupled system of real-valued PDEs:

(8) $$\frac{\partial v}{\partial t}=\frac{1}{\text{2}}\frac{{\partial }^{2}u}{{\partial \theta }^2}-\kappa \left(1-\cos \theta \right)u,$$

(9) $$\frac{\partial u}{\partial t}=-\frac{1}{\text{2}}\frac{{\partial }^{2}v}{{\partial \theta }^2}+\kappa \left(1-\cos \theta \right)v.$$

This decomposition forms the very structure directly implemented into the PINN loss function: the network generates dual channels $(u,v)$, and the physical residual $ℛ$ is computed as the ${ℒ}^2$ norm of (Equations 8)–(9) evaluated at collocation points. The Hermiticity of the Hamiltonian operator ensures that this coupled system preserves the norm ${\Vert u\Vert }^2+{\Vert v\Vert }^2$ under exact evolution, providing an intrinsic consistency check for the learned representation.

The stationary states (eigenstates) abide by the Time-Independent Schrödinger Equation (TISE):

(10) $$-\frac{1}{\text{2}}\frac{d^2{\psi }_n}{{d\theta }^2}+\kappa \left(1-\cos \theta \right){\psi }_n=E_n{\psi }_n.$$

Applying the variable substitution $z=\theta /2$ into the TISE and rearranging the terms reduces this equation directly to the canonical form of the Mathieu Equation [¹⁴]:

(11) $$\frac{d^2\psi }{{dz}^2}+\left[a-2q\cos \left(2z\right)\right]\psi =0,$$

where the Mathieu parameters $(a,q)$ map exactly to the physical constants as $q=-4\kappa$ and $a=8\left(E_n-\kappa \right)$. The $2\pi$ periodicity requirement on $\theta$ translates into a $\pi$ periodic boundary on $z$, restricting the allowable solutions to the so-called Mathieu functions: the cosine-elliptic $c{e}_n(z,q)$ and sine-elliptic $s{e}_n(z,q)$ functions, which act as the trigonometric analogs for non-linear periodic potentials. In this paper, we focus on the ground state, described by the even function $c{e}_0(\theta /2,-4\kappa )$, whose eigenvalue $a_0\left(q\right)$ is obtained numerically via scipy.special.mathieu_a. This mathematical object provides an exact analytic benchmark and serves as the initial condition for the PINN. The potential profile and the initial probability distribution are displayed in Figure 1.

Figure 1
Dimensionless cosine potential $V\left(\theta \right)=\kappa \left(1-\cos \theta \right)$ with $\kappa =5.0$. Horizontal axis: angular displacement $\theta \in [-\pi ,\pi ]$ (radians); vertical axis: energy $V$ (units of ${ℰ}_r={\hslash }^2/I$). The solid curve (blue) illustrates the potential landscape; the filled region (orange) represents the initial probability distribution ${\left|{\psi }_0\right|}^2={\left|c{e}_0(\theta /2,q)\right|}^2$ of the Mathieu ground state, centered at $\theta =0$. The dashed horizontal line (red) indicates the classical barrier threshold $V_{\max }=2\kappa$, above which exponential penetration (quantum tunneling) occurs.

2.1.4. Topological quantization and the operator domain

Analytically, the Hilbert space is formulated upon the quotient space $ℝ/2\pi \mathbb{Z}$. The self-adjointness of the momentum operator ${\widehat{p}}_{\theta }=-i{\partial }_{\theta }$ explicitly demands that the domain $𝒟\left({\widehat{p}}_{\theta }\right)$ be restricted to strictly periodic functions:

(12) $$𝒟\left({\widehat{p}}_{\theta }\right)=\{\psi \in {ℒ}^2[0,2\pi ]:\psi \left(0\right)=\psi \left(2\pi \right)\}.$$

In standard Deep Learning empirical frameworks (Soft Constraints), this boundary condition is circumvented utilizing an iterative boundary penalty term ${ℒ}_{BC}={\left|\psi \left(-\pi \right)-\psi \left(\pi \right)\right|}^2+{\left|{\psi }^{\prime }\left(-\pi \right)-{\psi }^{\prime }\left(\pi \right)\right|}^2$. However, as observed in recent architecture assessments, the topological gradient ${\nabla }_{\theta }{ℒ}_{BC}$ vanishes almost everywhere except precisely at the boundary $\partial \Omega$, precipitating the “vanishing gradient on manifolds” pathology [⁸].

By immersing the coordinate $\theta$ into the unit circle via $𝒯\left(\theta \right)=(\cos \theta ,\sin \theta )$, we construct a composite map $\widetilde{\text{Ψ}}=𝒩\circ 𝒯$. The chain rule assures strict periodicity by construction:

(13) $$\begin{array}{l}\widetilde{\text{Ψ}}\left(\theta +2\pi \right)=𝒩(\cos \left(\theta +2\pi \right),\sin \left(\theta +2\pi \right))\\ =𝒩(\cos \theta ,\sin \theta )=\widetilde{\text{Ψ}}\left(\theta \right).\end{array}$$

This operation effectively roots the neural network onto the intrinsic manifold $S^1$, eschewing the need for Lagrange multipliers or penalty terms and guaranteeing that the solution remains strictly bound within $𝒟\left(\widehat{H}\right)$.

2.2. The topological embedding(hard constraints)

We circumvent this pathology by establishing an isometric embedding of the compact Riemannian manifold $(S^1,g_{S^1})$ directly into the ambient Euclidean space $({ℝ}^2,g_{\text{Eucl}})$ via the smooth mapping $𝒯:S^1\hookrightarrow {ℝ}^2$, quantified as $𝒯:\theta \mapsto [\cos \left(\theta \right),\sin \left(\theta \right)]$. This embedding inherently preserves the intrinsic metric of $S^1$, ensuring the periodic constraint $\text{Ψ}\left(\theta \right)=\text{Ψ}\left(\theta +2\pi \right)$ is fulfilled by the explicit topology of the domain rather than by extrinsic loss penalties. The input tensor injected into the network is therefore:

(14) $${\mathbf{x}}_{\text{input}}=[t,\cos \left(\theta \right),\sin \left(\theta \right)].$$

Differentiability of the neural ansatz is perpetually upheld by the chain rule. For the target embedding $\mathbf{u}=(\cos \theta ,\sin \theta )$, the spatial derivative of the network’s final output $\text{Ψ}\left(\mathbf{u}\right)$ concerning the raw physical coordinate $\theta$ is:

(15) $$\frac{\partial \text{Ψ}}{\partial \theta }=\frac{\partial \text{Ψ}}{\partial u}\frac{\partial u}{\partial \theta }+\frac{\partial \text{Ψ}}{\partial v}\frac{\partial v}{\partial \theta }=-\sin \theta \frac{\partial \text{Ψ}}{\partial u}+\cos \theta \frac{\partial \text{Ψ}}{\partial v}.$$

This mechanism entitles the network to learn within the immersed sub-space $(u,v)$ while autonomously maintaining periodic boundary conditions $\text{Ψ}\left(\theta \right)=\text{Ψ}\left(\theta +2\pi \right)$ alongside smooth differentiability throughout the boundary cut.

Essentially, this geometric mapping perceives the pendulum not as a linear localized particle bounded by walls, but as an invariant point navigating the unit circle enclosed within 2D Euclidean space. The neural network $𝒩$ effectively acts as a function $𝒩(\cos \theta ,\sin \theta )$. Considering that $\cos \left(-\pi \right)=\cos \left(\pi \right)=-1$ and $\sin \left(-\pi \right)=\sin \left(\pi \right)=0$, opposing frontiers become algebraically identical. Furthermore, spatial derivatives $\frac{\partial \text{Ψ}}{\partial \theta }$ remain reliably sound due to the $C^{\infty }$-continuity characterizing the mapping $𝒯$ across $S^1$. The resulting $\text{Ψ}$ embodies periodic and continuous traits securely by construction, fulfilling manifold requirements immune to unstable penalties [⁸].

2.3. The topological neural network

As architecturally modeled in the visual schematic of Figure 2, the proposed routing apparatus processes independent scalar tensors for both time $t$ and angular space $\theta$. Nonetheless, to shield the system against $C^{\infty }$-discontinuity across cyclic transitions $\text{Ψ}(-\pi ,t)=\text{Ψ}(\pi ,t)$, spatial coordinates are fundamentally transformed preceding the Multilayer Perceptron (MLP) within the Hard Constraint block (Figure 3). The mathematical protocol $𝒯:\theta \mapsto (\cos \theta ,\sin \theta )$ successfully folds 1D topology onto the $S^1$ sphere.

Figure 2
Proposed PINN architecture designated for the Quantum Pendulum scenario. Coordinate boundary $\theta$ is rigidly mapped to $S^1\subset {ℝ}^2$ manifold utilizing $𝒯:\theta \mapsto [\cos \left(\theta \right),\sin \left(\theta \right)]$. This distinct Hard Constraint protocol mechanically assures $\text{Ψ}\left(-\pi \right)=\text{Ψ}\left(\pi \right)$ by design, subverting gradient pathology instabilities originally discovered by Dong & Ni (2021) [⁸].

Figure 3
Detailed schematic representing the Trigonometric Coordinate Embedding layer. The angular scalar $\theta$ translates explicitly onto metric positions $\cos \theta$ alongside $\sin \theta$, forcing unbroken topological continuity leading into the interconnected hidden networks.

Following the geometric manifold embedding, multivariate inputs $[t,\cos \theta ,\sin \theta ]$ flush through the fully-connected MLP outfitted with hyperbolic tangent activation functions. Crucially, given the intrinsically complex nature of solutions to the Schrödinger equation, predictions mandate dual-channel representation. Thus, the terminal dense projection partitions outputs strictly into real scalars $u(\theta ,t)$ and $v(\theta ,t)$. These parameters linearly recombine under the algebraic identity $\text{Ψ}(\theta ,t)=u(\theta ,t)+iv(\theta ,t)$, supplying the required wavefunction configuration necessary for measuring the physical loss residual ${ℒ}_{physics}$.

To empirically validate this proposition, we formulated a head-to-head convergence assessment pitting the Hard Constraint network (lacking boundary penalization entirely) against an identical Soft constraint baseline framework (mandating strong ${\lambda }_{BC}=100$ penalties). The models were concurrently trained employing $N_f=20,000$ internal collocation points under a regimen of $5,000$ L-BFGS epochs. As evident within Figure 4, implementing rigid embeddings extinguishes recurrent instability spikes completely, securing a cumulative loss metric $1.43\times$ lower than standard Soft paradigms ($1.24\times {10}^{-2}$ contrasted against $1.78\times {10}^{-2}$).

Figure 4
Convergence efficiency comparison: Soft Constraints (empirical boundary penalty, denoted in red) vs. Hard Constraints (topological trigonometric immersion, denoted in blue). While the Soft framework experiences cyclic instabilities plus elevated final residual floors, our rigidly formulated PINN decays symmetrically while yielding a definitive $1.43\times$ mathematical loss advantage.

2.4. Optimization landscape

The network minimizes the composite loss function $ℒ$, defined explicitly by the weighted sum of the physical residual (${ℒ}_{phys}$) and the initial condition penalty (${ℒ}_{IC}$):

(16) $$ℒ={ℒ}_{phys}+{\lambda }_{IC}{ℒ}_{IC},$$

where the terms are expressed individually as summations over the sampled collocation points:

(17) $${ℒ}_{phys}=\frac{1}{N_f}\sum _{i=1}^{N_f}[{\left({\frac{\partial v}{\partial t}|}_i+{\widehat{H}u|}_i\right)}^2+{\left({\frac{\partial u}{\partial t}|}_i-{\widehat{H}v|}_i\right)}^2],$$

(18) $$\begin{array}{l}{ℒ}_{IC}=\frac{1}{N_0}{\displaystyle \sum _{j=1}^{N_0}}[{(u(t_0,{\theta }_j)-u_0^{exact}\left({\theta }_j\right))}^2\\ +{(v(t_0,{\theta }_j)-v_0^{exact}\left({\theta }_j\right))}^2].\end{array}$$

Here, $\widehat{H}\varphi =-\frac{1}{\text{2}}\frac{{\partial }^2\varphi }{{\partial \theta }^2}+\kappa \left(1-\cos \theta \right)\varphi$ is the dimensionless Hamiltonian operator, $N_f=20,000$ is the number of spatiotemporal collocation points distributed in the domain $[-\pi ,\pi ]\times [0,T_{max}]$, $N_0=2,048$ is the number of spatial points for the initial condition at $t_0=0$, and ${\lambda }_{IC}=10.0$ is the empirical regularization weight. Note that no boundary penalty term ${ℒ}_{BC}$ is included in the loss, as periodicity is rigidly guaranteed by the embedding. We employ a two-stage training strategy:

1. Global Search (Adam): 10,000 epochs with a learning rate of ${10}^{-3}$ to explore the rugged loss landscape.

2. Fine-Tuning (L-BFGS): A quasi-Newton method to refine the solution with high precision ($𝒪\left({10}^{-4}\right)$).

The convergence history under this hybrid regime is presented in Figure 4(b). We emphasize that using ${\lambda }_{IC}=10.0$ to weight the initial condition proved crucial to ensure the wavepacket tracks the initial state at $t=0$ before the L-BFGS phase begins.

For reproducibility, the specific hyperparameters used in this study are detailed in Table 2.

Table 2
Hyperparameters and Computational Configurations.

Parameter	Value
Network Architecture	MLP (Feed-forward)
Hidden Layers	4
Neurons per Layer	50
Trainable Parameters	$\approx$ 10,500
Activation Function	$\tanh$
Optimizer 1 (Global)	Adam ($\alpha ={10}^{-3}$, 10k epochs)
Optimizer 2 (Local)	L-BFGS (Quasi-Newton)
Potential Strength ($\kappa$)	5.0
Final Loss (Adam)	$4.01\times {10}^{-3}$
Number of Collocation	20,000
Points ($N_f$)
Training Time	$\approx$ 400 s (RTX 3050 GPU)

3. Results and Discussion

The system is initialized in the Mathieu ground state. The evolution is shown in Figure 5. Unlike the Harmonic Oscillator, where the ground state is trivial, the cosine potential induces nonlinear dispersion. We validate the network against the exact analytical solution ${\text{Ψ}}_{analytic}$ derived from the Mathieu functions [¹⁴], rather than a discrete numerical integration, ensuring the highest benchmark rigor. The topological consistency of the solution on the cylindrical manifold $S^1\times ℛ$ is visually confirmed in Figure 6, where the seamless continuity at the periodic boundary $\text{Ψ}\left(-\pi \right)=\text{Ψ}\left(\pi \right)$ validates the hard-constraint embedding.

Figure 5
Spatiotemporal evolution of the probability density ${\left|\text{Ψ}(\theta ,t)\right|}^2$. The left panel shows the analytical Mathieu baseline, and the right panel displays the neural prediction in the non-linear regime. As the system is initialized in the Mathieu ground state, the exact solution must be a stationary state: ${\left|\text{Ψ}(\theta ,t)\right|}^2={\left|{\psi }_0\left(\theta \right)\right|}^2$. The minor width modulations visible in the solver output are residual coherence fluctuations, acting as a thermometer for strict phase conservation in the PINN simulation.

Figure 6
Visualization of topological evolution on $S^1\times ℝ$. The continuous cylindrical surface confirms that the neural solution satisfies $\text{Ψ}\left(-\pi \right)=\text{Ψ}\left(\pi \right)$ and its derivatives continuously, validating the hard-constraint trigonometric embedding without requiring boundary penalty terms.

3.1. Quantum coherence and tunneling

To confirm the presence of tunneling, we perform a logarithmic scale analysis (Figure 7). A pedagogical detail that frequently confuses students of quantum mechanics is the possibility of the local probability density ${\left|\text{Ψ}(\theta ,t)\right|}^2$ assuming values strictly greater than $1.0$ (as observed in the central region of our plots). As dictated by quantum formalism, the normalization condition requires the integral over the configuration space to be unitary (${\int }_{-\pi }^{\pi }{\left|\text{Ψ}\right|}^2\mathit{d}\theta =1$). Since the domain has an arc length of $2\pi$, the mean uniform density is $1/\left(2\pi \right)\approx 0.159$. Consequently, for states strongly localized like the Mathieu ground state at the bottom of the well, the local density at the central peak reaches values near $1.2$, compensated by the decay in forbidden regions. The physical limit of probability being less than or equal to $1$ applies to the integrated probability over a subinterval, not to the local density, which can be arbitrarily high (e.g., a Dirac delta distribution). The exponential tails in the classically forbidden region are the signature of macroscopic quantum tunneling: the linear decay of $\log \left({\left|\text{Ψ}\right|}^2\right)$ corresponds to the penetration depth $\xi \propto 1/\sqrt{\kappa -E}$, a direct manifestation of the WKB regime where the wavefunction “leaks” through the potential barrier. A standard linear plot would dismiss these signatures as noise.

Figure 7
Logarithmic scale analysis of the wavefunction. The overlay represents the theoretical WKB decay $\text{Ψ}\propto \exp \left(-\int \sqrt{2\left(V-E\right)}\right)$, confirming the PINN captures the semiclassical attenuation in the forbidden region without explicit boundary enforcement. The linear regression provides the decay rate ${\xi }_{\text{PINN}}\approx 0.304$ against the approximate ${\xi }_{\text{WKB}}\approx 0.357$.

To quantitatively validate the tunneling tails, we compared the decay rate of the PINN solution with the theoretical WKB penetration coefficient. In the classically forbidden region ($\left|\theta \right|>{\theta }_c$, where ${\theta }_c=\arccos \left(1-E_0/\kappa \right)$ is the classical turning point), the WKB approximation predicts an exponential decay ${\left|\text{Ψ}\left(\theta \right)\right|}^2\propto \exp \left(-2\theta /{\xi }_{\text{WKB}}\right)$ with a characteristic penetration depth:

(19) $${\xi }_{\text{WKB}}=\frac{1}{\sqrt{2\left(\kappa -E_0\right)}}.$$

For the parameters of this study ($\kappa =5.0$, $E_0\approx 1.0858$), the theoretical value is ${\xi }_{\text{WKB}}\approx 0.357$. Performing a linear regression of $\log \left({\left|{\text{Ψ}}_{\text{PINN}}\right|}^2\right)$ in the classically forbidden region ($\theta \in [1.0,2.5]$), we extracted the characteristic penetration depth predicted by the network as ${\xi }_{\text{PINN}}\approx 0.304$. The relative discrepancy compared to theory is $14.85\%$, an acceptable qualitative agreement that validates the PINN’s capability to resolve evanescent wavefunction tails without explicit boundary imposition at the barrier. (It should be noted that the ${\xi }_{\text{WKB}}$ metric obtained above constitutes a rough approximation assuming $V\approx \kappa$ constant in the barrier. The exact WKB integral over the variable potential $\kappa \left(1-\cos \theta \right)$ would justify the residual discrepancy observed).

To locally quantify the solver’s accuracy, we analyze the spatiotemporal absolute error distribution presented in panel (c) of Figure 5. As observed, the absolute error presents a local maximum located at the origin $\theta =0$ at $t=0$. This behavior is physically coherent with the system’s initialization in the Mathieu ground state, which is strongly concentrated at the bottom of the well. Small network deviations in imposing the initial condition result in slightly larger absolute residuals exactly where the density is maximal. As the evolution proceeds, the error diffuses, exhibiting low-amplitude fluctuations concentrated along the central channel and near the potential walls ($\theta \approx \pm 2$), remaining strictly below the upper bound of $1.5\times {10}^{-2}$ throughout the integration domain.

3.2. Probability current residual diagnostics

Beyond density profiles, the dynamics on $S^1$ can be evaluated by mapping the probability current $j(\theta ,t)=\text{Im}\left({\text{Ψ}}^{\ast }{\partial }_{\theta }\text{Ψ}\right)$, which describes the spatial flow of probability. Figure 8 presents a spatiotemporal map of $j(\theta ,t)$. For the stationary ground state modeled here, the theoretical probability current is identically zero ($j_{\text{exact}}=0$), as stationary states carry no net flux. The alternating flow patterns observed in the PINN output therefore constitute numerical phase artifacts, originating from the imperfect cancellation of spatial gradients. The strictly bounded magnitude of these residual currents ($\left|j\right|<𝒪\left({10}^{-2}\right)$) serves as a rigorous diagnostic of the network’s phase coherence.

Figure 8
Spatiotemporal map of the residual probability current $j(\theta ,t)$. As the physical expectation for the stationary state is $j_{\text{exact}}=0$, the bounded fluctuations ($\left|j\right|<𝒪\left({10}^{-2}\right)$) diagnose the network’s phase coherence and its temporal cancellation capabilities.

3.3. Physical consistency checks

PINNs are not symplectic integrators; they do not conserve invariants by construction. We must therefore verify them a posteriori.

It is worth noting that no explicit penalty term for norm conservation (e.g., ${ℒ}_{norm}={\left|1-⟨\text{Ψ}|\text{Ψ}⟩\right|}^2$) was added to the loss function. The observed stability of the norm ($>0.99$) is an empirical consequence of minimizing the residual of the Hermitian Hamiltonian. This suggests that the “Hard Constraint” embedding not only regularizes the spatial topology but also stabilizes the unitary evolution, a property frequently requiring symplectic integrators in classical schemes.

Quantitative results are summarized in Table 3.

Table 3
Quantitative Metrics for Physical Validation.

Metric	Value
Relative L2 Error (Global)	$1.42\times {10}^{-2}$
L2 Error (Initial Condition)	$2.0\times {10}^{-6}$
Mean Norm $⟨𝒩⟩$	$>0.999$
Mean Energy $⟨E⟩$	$1.0858\pm 0.005$

The temporal audit presented in Figure 9 attests that both conservation laws are maintained with extreme precision by the network. It is important to highlight that no explicit penalty for norm conservation was added to the cost function. Because the temporal evolution equations of quantum mechanics are governed by a self-adjoint (Hermitian) Hamiltonian operator, unitarity and total energy conservation are intrinsic mathematical properties of the differential equation. The numerical stability observed for the norm ($>0.999$, panel a) and total energy ($⟨H⟩=1.0858\pm 0.005$, panel b) emerges naturally from minimizing the Schrödinger equation residual under the rigid periodicity constraint, confirming the PINN captures the unitary symmetry of the quantum propagator without ad-hoc additives.

Figure 9
Temporal audit of the PINN solver’s physical conservation laws. (a) Conservation of the wavefunction probability norm. The stability around 1.0 attests to the learned implicit unitarity. (b) Conservation of the Hamiltonian expectation value (total energy) of the pendulum $⟨H⟩\approx 1.0858$, confirming the dynamics are conservative.

3.4. Phase space analysis

Finally, the Ehrenfest dynamics (Figure 10) and the Momentum space (Figure 11) show the complementarity of the system. As position uncertainty $\text{Δ}\theta$ grows (delocalization), the momentum distribution $\text{Φ}\left(p\right)$ must also evolve to satisfy the Heisenberg Uncertainty Principle for a particle on a circle. The residual drift in the expectation values $\left(\theta \right)$ and $\left(p\right)$ is further visualized in the pseudo-empirical phase portrait (Figure 12), where the confined limit cycle confirms the numerical stability of the Hard Constraint solver.

Figure 10
Stationary State Verification. Left panels: Time series of $⟨\theta ⟩$ and $⟨p⟩$ showing tracking against the exact analytical baseline (where expectations are rigorously constant). Right panels: Numerical absolute error, revealing the PINN maintains the deviation bound below $𝒪\left({10}^{-2}\right)$ throughout the computational cycle.

Figure 11
Momentum space analysis. The spectral content of the wavefunction evolves, maintaining the phase space volume.

Figure 12
Bounding residual drift in the pseudo-empirical Phase Space $⟨\theta ⟩\times ⟨p⟩$. For an exact ground state, this diagram should constitute a singular point at the origin $(0,0)$. The observed elliptical loop is a numerical limit cycle (an artifact of the continuous PINN gradient minimization) rather than an equivalent macroscopic quantum orbit.

3.5. Spatiotemporal convergence andspectral bias

To demonstrate global physical commitment, the evolution of the Heisenberg Uncertainty Principle was mapped for the closed domain. On the circle $S^1$, conventional Euclidean variance $\text{Δ}{\theta }^2$ is physically flawed due to polar discontinuity, requiring the rigor of the Holevo Variance defined as $V_H\left(\theta \right)=1-{\left|⟨e^{i\theta }⟩\right|}^2$. The uncertainty product of our network (Figure 13) attests that the imposed constraint captures not only primary quantities but regulates the Holevo metric $\sqrt{V_H}\text{Δ}p\ge 0.5$ throughout the solver, avoiding the undue statistical deflation present in non-physical partial sub-solutions.

Figure 13
Heisenberg Uncertainty Audit. The use of the Holevo variance product ($\sqrt{V_H}\text{Δ}p$) shows the solution physically maintaining distance from the quantum limit (0.5) in empirical fluctuations, unlike the simplistic and inadequate use of the Euclidean proxy ($\text{Δ}\theta$) on $S^1$.

Furthermore, we analyzed the Spectral Bias by taking the Fast Fourier Transform (FFT) of the Schrödinger residual (Figure 14). The residual spectrum decays exponentially at higher frequencies, confirming the network accurately captured the fine-scale details of the dynamics and is not suffering from high-frequency noise typical of poorly converged mesh methods.

Figure 14
Spectral Bias Audit via FFT of the Schrödinger residual. Left panel: spatiotemporal distribution of the residual. Right panel: mean frequency spectrum. The absence of high-frequency noise peaks indicates a smooth, well-converged approximation of the wavefunction.

The norm drift observed beyond the training horizon ($t>T_{\text{train}}$, Figure 15) is an expected consequence of uniform temporal allocation. In our formulation, the physical loss ${ℒ}_{\text{physics}}$ is evaluated uniformly across all collocation points in the interval $[0,T]$, effectively treating temporal evolution as a spatial regression problem. Because the network minimizes a global residual surface rather than advancing sequentially like explicit solvers, initial spatial errors inevitably diffuse to later epochs, fueling cumulative norm degradation at $t>T_{\text{train}}$. Although advanced formulations like Causal Training [¹⁵] have been proposed to enforce sequential adaptive weighting, the extrapolation error bounded at $5\%$ observed in our results (without such adaptive interventions) constitutes a robust baseline against which future causally-aware implementations can be compared.

Figure 15
Long-Term Stability Audit. The norm conservation metric $\int {\left|\text{Ψ}\right|}^2\mathit{d}\theta$ is tracked beyond the training horizon ($t>T_{train}$). The error remains bounded within a 5% margin for extrapolation, evidencing that the topological embedding regularizes the solution against overfitting.

The pseudo-orbital geometric constriction plotted in Figure 12 provides visual confirmation of latent numerical stability. As a stationary state, the Hamiltonian formalism predicts strict coordination resting at the origin $(0,0)$. Thus, it is assertively outlined that the plotted curve isolates numerical noise under the manifest appearance of a loop. However, this finding is positive: a soft constraint would spiral infinitely, whereas the Hard Embedding-based PINN ties these errors into a confined limit cycle, preventing artificial macroscopic escape from expected values under long sampling.

3.6. Generalization out of equilibrium:displaced Gaussian wavepacket

As a test of methodological robustness requested in peer review, we evaluate the PINN solver under a dynamic, out-of-equilibrium initial condition represented by a displaced Gaussian wavepacket:

(20) $$\text{Ψ}(\theta ,0)=\frac{1}{{\left(\pi {\sigma }^2\right)}^{1/4}}\exp \left(-\frac{{\left(\theta -{\theta }_d\right)}^2}{2{\sigma }^2}\right),$$

where the initial wavepacket center is defined at ${\theta }_d=1.0$ and the width is $\sigma =0.5$. This non-stationary state exhibits complex oscillatory (breathing) and dispersion dynamics in the cosine potential well, colliding with and reflecting off the potential walls. We train the Hard-Constrained PINN under the same hyperparameters as before and validate the results against a high-fidelity numerical simulation obtained using the Split-Step Fourier Method (SSFM) as the ground truth.

Due to the asymptotic nature of the Gaussian wavepacket, it does not satisfy strict periodicity at the topological boundaries ($\text{Ψ}\left(-\pi \right)\ne \text{Ψ}\left(\pi \right)$). However, for $\sigma =0.5$, the boundary tails are of order $𝒪\left({10}^{-4}\right)$, making this approximation numerically acceptable for the out-of-equilibrium study in question.

The results are compiled in Figure 16. The probability density heatmaps of the exact solution (panel a) and the PINN prediction (panel b) show perfect visual agreement. The spatiotemporal local absolute error (panel c) remains strictly confined below $3.5\times {10}^{-2}$ throughout the integration interval. For a quantitative detailing, panel (d) shows spatial snapshots at $t=0.0$, $2.5$, and $5.0$ s, demonstrating that the network prediction (blue circles) tracks the exact solution’s evolution and phase oscillations (red dashed line) with extremely high fidelity. This robust generalization test independently validates the flexibility of the neural solver under states far from quantum equilibrium.

Figure 16
Validation of out-of-equilibrium generalization under a displaced Gaussian wavepacket (${\theta }_d=1.0$, $\sigma =0.5$). (a) Exact evolution obtained via the Split-Step Fourier Method (SSFM). (b) Spatiotemporal evolution predicted by the proposed Hard PINN. (c) Spatiotemporal absolute error map, demonstrating error strictly bounded below $3.5\times {10}^{-2}$. (d) Comparative spatial profiles at $t=0.0$, $2.5$, and $5.0$ s. The PINN accurately captures the oscillations, breathing, and reflection effects.

4. Conclusion

This work presented a topological framework designed to solve the Time-Dependent Schrödinger Equation on the circle ($S^1$) leveraging Physics-Informed Neural Networks. Diverging from standard approaches that treat periodic limits via soft penalty terms [¹, ⁴], our method immerses the coordinate space directly into a non-Euclidean manifold via $𝒯:\theta \mapsto (\cos \theta ,\sin \theta )$. This “Hard Constraint” architecture imposes strict $C^{\infty }$ continuity and periodicity by construction, eliminating the “vanishing gradient on manifolds” problem identified in recent geometric deep learning literature [⁸].

4.1. Comparative analysis

Our results demonstrate a significant methodological advancement over early neural solvers. Lagaris et al. (1998) [¹] originally proposed solving differential equations using test functions with satisfactory boundary conditions. While effective for simple Dirichlet boundaries, their approach struggles with periodic conditions where boundary values are unknown a priori. Recent developments in PINNs [⁴] typically address this with ${ℒ}_{BC}$ loss terms. However, as shown in our Stability Audit (Figure 15), such soft constraints permit norm drift (${\Vert \text{Ψ}\Vert }^2\ne 1$) over long integration times. In contrast, our topological embedding maintains the ground state trajectory in phase space (Figure 12) with high fidelity, preventing the “spiraling” often seen in unconstrained solvers. By eliminating the ${ℒ}_{BC}$ term, the Hard Constraint significantly simplifies the optimization landscape, fundamentally explaining its superior convergence over Soft Constraint alternatives.

4.2. Physical implications

The successful simulation of the Quantum Pendulum has broader implications for macroscopic quantum phenomena. The observed stability of the wavefunction on the $S^1\times ℝ$ manifold suggests this architecture is particularly suited for modeling Superconducting Qubits, such as the Cooper Pair Box and Transmon. These devices are governed by a Hamiltonian mathematically isomorphic to the pendulum ($H\sim E_C{\widehat{n}}^2-E_J\cos \widehat{\varphi }$), where the phase variable $\varphi$ is compact on $S^1$[¹⁶]. Consequently, the topological deep learning framework presented here is directly applicable to simulating superconducting quantum circuits. Furthermore, the ability to capture tunneling tails (Figure 7) without mesh refinement offers an advantage over Finite Element Methods (FEM) in high-dimensional configuration spaces.

4.3. Limitations and future work

Despite the advances obtained with the Hard Constraint embedding, the present solver exhibits limitations that outline directions for future investigations:

1. Unitary Conservation and Gauge Symmetry: Unitarity and $U\left(1\right)$ temporal phase symmetry are still softly enforced via the physical residual. The introduction of neural architectures that conserve symplectic invariants or that incorporate dynamic, self-adjusting Lagrange multipliers constitutes a promising avenue to guarantee strict energy and norm conservation by construction in asymptotic times.

2. Scalability and Strong Coupling: The study of strong coupling potentials (high $\kappa$ regime), characteristic of chaotic transitions and turbulent dynamics in the quantum phase space, requires the deployment of adaptive sampling (e.g., Residual-Based Adaptive Distribution) to concentrate collocation points on highly oscillatory phase fronts.

3. Multidimensional Extensions and Non-Euclidean Spaces: The generalization of this formalism to high-dimensional manifolds, such as a three-dimensional rigid rotor immersed in the rotation group $SO\left(3\right)$ or the Bloch sphere $S^2$, will enable mesh-free simulations of entangled multi-qubit systems under dissipative coupling (Caldeira-Leggett model), overcoming the dimensionality bottleneck that renders classical grid methods intractable.

Supplementary Material – Reproducibility Code and Computational Notebooks

The complete implementation code and pedagogical Jupyter notebooks referenced in this article are publicly available at the repository listed under Data Availability. The supplementary material includes:

• PINN_Quantum_Pendulum_Solver.ipynb: Interactive notebook demonstrating the core PINN solver with topological embedding on $S^1$.

• Out_of_Equilibrium_Robustness_Test.ipynb: Validation of the displaced Gaussian wavepacket against the Split-Step Fourier Method.

• Soft_vs_Hard_Constraints_Comparison.ipynb: Comparative experiment between soft boundary penalization and hard topological embedding.

• pêndulo_quântico_schrodinger.py: Architecture definition of the Hard Constraint PINN with the Trigonometric Coordinate Embedding layer.

• generate_ground_truth.py: Exact analytical benchmark computation via Mathieu functions using scipy.special.

• soft_vs_hard_experiment.py: Training loops and convergence comparison scripts.

• generate_out_of_equilibrium.py: Displaced Gaussian wavepacket training and validation script.

• Pre-trained assets:quantum_ground_truth.npz, quantum_pinn_model.pth, and quantum_pinn_gaussian.pth, enabling immediate evaluation without retraining.

Dependencies: Python 3.10+, PyTorch, SciPy, NumPy, and Matplotlib.

Acknowledgments

The authors thank the Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE), Campus Fortaleza, for the institutional support provided for this research. The authors are also grateful to their professors for the academic guidance and scientific mentorship throughout the development of this work.

Data Availability

The entire dataset supporting the results of this study, including source code, pre-trained neural network weights, analytical reference solutions, and interactive Jupyter notebooks, is publicly available in a data repository:

• Repository name: GitHub

• URL: https://github.com/jjcarlosb-hue/quantum-pendulum-pinn

• Deposit date: July 2026

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Edited by

• Editor-in-Chief:
  Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  07 Sept 2026
• Date of issue
  2026

History

• Received
  01 Apr 2026
• Reviewed
  22 June 2026
• Accepted
  29 June 2026