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arxiv: 2502.05621 · v2 · submitted 2025-02-08 · 🪐 quant-ph

A Pedagogical Framework for Physics-Informed Machine Learning: From Classical Pendulum to Quantum Anharmonic Oscillator Using PyTorch on Modern GPU Hardware

Enis Yazici This is my paper

Pith reviewed 2026-05-23 03:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords physics-informed neural networksPINNnonlinear pendulumquantum anharmonic oscillatorPyTorchGPU accelerationpedagogical frameworkmachine learning for physics
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The pith

A five-module curriculum teaches the transition from data-driven neural networks to physics-informed models on pendulum and quantum oscillator systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a structured pedagogical framework that implements and compares five neural network architectures across two physical systems of increasing complexity. It begins with data-driven models for a driven damped nonlinear pendulum and advances to physics-informed neural networks for a one-dimensional quantum anharmonic oscillator, all coded in PyTorch and run on modern GPU hardware. Quantitative results include specific error values for each approach along with measured CPU-to-GPU speedups. A sympathetic reader would care because the work supplies ready-to-use course materials that move students from purely data-driven thinking to formulations that embed differential equations and boundary conditions directly into the loss function.

Core claim

The authors present a five-module framework that deploys an ANN, a 1D CNN, an LSTM, and two separate PINNs on the pendulum and quantum oscillator problems. Data-driven models reach mean absolute errors of 1.3×10^{-2} rad on the pendulum and 4.4×10^{-5} a.u. on the quantum system, while a curriculum-trained pendulum PINN attains 3.1×10^{-2} rad using only collocation points. The same implementations yield GPU speedups between 1.2× and 24.6× depending on architecture size, and the materials are delivered as self-contained Jupyter notebooks with embedded reflection questions.

What carries the argument

The five-module pedagogical framework that sequences data-driven architectures (ANN, CNN, LSTM) before physics-informed neural networks (PINNs) on the driven damped pendulum and quantum anharmonic oscillator.

If this is right

  • Students completing the modules can directly compare how embedding the equations of motion into the loss changes accuracy and data requirements.
  • The measured speedups indicate that GPU acceleration becomes worthwhile once model size or sequence length increases beyond small feed-forward networks.
  • The progression from pendulum to quantum oscillator supplies a concrete path for extending the same curriculum to other classical-to-quantum pairs.
  • The packaged notebooks allow a graduate course to run the full set of experiments without external data sources beyond the physical models themselves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The gap between data-driven and physics-informed errors on the pendulum suggests that, when abundant trajectory data exist, pure supervised learning may remain preferable unless extrapolation beyond the training domain is required.
  • Curriculum training, shown here to help the PINN, could be tested on other differential-equation problems where standard PINN training stalls.
  • The framework's emphasis on modern GPU hardware implies that similar courses could incorporate larger quantum systems or higher-dimensional oscillators without changing the pedagogical structure.

Load-bearing premise

The code correctly implements the governing differential equations, damping, driving terms, and boundary conditions for both the classical pendulum and the quantum oscillator, and the reported error numbers accurately represent model performance.

What would settle it

An independent re-implementation of the curriculum-trained pendulum PINN that cannot reach a mean absolute error of 3.1×10^{-2} rad when trained solely on collocation points without additional labeled data would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2502.05621 by Enis Yazici.

Figure 1
Figure 1. Figure 1: Simulation of the pendulum with complex forces: angular displace [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: True vs. Predicted Angular Displacement for the Pendulum using a [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pendulum PINN: True vs. PINN-Predicted Angular Displacement. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quantum oscillator data: discretized potential and associated wave [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training and Validation Loss for the ANN (CNN) model for the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: True vs. Predicted Ground State Energy Levels using the ANN (CNN) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Training and Validation Loss for the LSTM model for the quantum [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: True vs. Predicted Ground State Energy Levels using the LSTM [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Quantum PINN: True vs. PINN-Predicted Wavefunction. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

We present a five-module pedagogical framework for teaching physics-informed machine learning (ML) through two progressively complex physical systems: a driven, damped nonlinear pendulum and a one-dimensional quantum anharmonic oscillator. Five model architectures are implemented and compared: a standard artificial neural network (ANN), a one-dimensional convolutional neural network (CNN), a long short-term memory (LSTM) network, and two physics-informed neural networks (PINNs) -- one per physical system. All models are implemented in PyTorch~2.9 and executed on an NVIDIA RTX~5090 GPU, making the framework directly applicable to modern deep learning laboratory courses. Quantitative benchmarks show that data-driven models achieve mean absolute errors of $1.3\times10^{-2}$~rad (pendulum ANN) and $4.4\times10^{-5}$~a.u.\ (quantum CNN), while the curriculum-trained pendulum PINN reaches an MAE of $3.1\times10^{-2}$~rad using only collocation points. A systematic CPU-vs-GPU benchmark reveals speedups ranging from $1.2\times$ (small ANN) to $24.6\times$ (LSTM), providing a concrete pedagogical demonstration of when GPU acceleration is -- and is not -- warranted. The framework is packaged as self-contained Jupyter notebooks designed for a graduate-level \emph{Deep Neural Networks for Physical Systems} course, with embedded reflection questions that guide students from data-driven thinking toward physics-constrained formulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a five-module pedagogical framework for physics-informed machine learning using a driven damped nonlinear pendulum and a one-dimensional quantum anharmonic oscillator. It implements and benchmarks five architectures (ANN, CNN, LSTM, and two PINNs) in PyTorch 2.9 on an NVIDIA RTX 5090 GPU, reporting MAEs of 1.3×10^{-2} rad (pendulum ANN), 4.4×10^{-5} a.u. (quantum CNN), and 3.1×10^{-2} rad (curriculum-trained pendulum PINN using only collocation points), along with CPU-GPU speedups from 1.2× to 24.6×. The work is delivered as self-contained Jupyter notebooks with reflection questions for a graduate course on deep neural networks for physical systems.

Significance. If the reported error metrics are reproducible and the physics constraints are correctly encoded, the framework supplies a concrete, progressive teaching sequence that demonstrates the shift from purely data-driven to physics-informed models and quantifies hardware acceleration trade-offs. The packaging as executable notebooks with embedded questions is a strength for classroom use.

major comments (1)
  1. [Abstract] Abstract: the central quantitative claim that the curriculum-trained pendulum PINN achieves an MAE of 3.1×10^{-2} rad using only collocation points is load-bearing for the pedagogical progression. Without the explicit residual loss term (including the sin(θ) nonlinearity, damping, and driving force) or the precise collocation-point enforcement of initial/boundary conditions, it is impossible to confirm that the reported error reflects genuine physics-informed training rather than an implementation artifact.
minor comments (1)
  1. The abstract states specific MAE values and speedups but provides no table or figure reference for the full set of benchmarks; adding a summary table of all five models would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit context around the central PINN claim. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central quantitative claim that the curriculum-trained pendulum PINN achieves an MAE of 3.1×10^{-2} rad using only collocation points is load-bearing for the pedagogical progression. Without the explicit residual loss term (including the sin(θ) nonlinearity, damping, and driving force) or the precise collocation-point enforcement of initial/boundary conditions, it is impossible to confirm that the reported error reflects genuine physics-informed training rather than an implementation artifact.

    Authors: The residual loss for the pendulum PINN is defined explicitly in Equation (5) of Section 3.2, consisting of the second-derivative term, the sin(θ) nonlinearity from the pendulum equation, the linear damping term γ dθ/dt, and the driving term F cos(ωt), all evaluated at collocation points. Initial conditions are enforced through a separate data-loss term on the initial segment of the trajectory, while the collocation points enforce the differential residual throughout the interior. We acknowledge that the abstract is too terse to convey this structure and will revise it to include a concise parenthetical reference to the residual formulation (e.g., “via a residual loss that incorporates the sin(θ) nonlinearity, damping, and driving force”). This change makes the claim self-contained while preserving the reported MAE value. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical benchmarking of implementations with no derivation chain

full rationale

The paper is a pedagogical implementation and benchmarking exercise. It reports empirical MAE values obtained by training PyTorch models (ANN, CNN, LSTM, PINNs) on two physical systems. No first-principles derivations, uniqueness theorems, or predictions are claimed that could reduce to fitted inputs or self-citations by construction. The reported errors are direct outputs of model training runs, not algebraic identities. The work is self-contained against external benchmarks (GPU timings, error metrics) with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard neural-network training assumptions and the known differential equations for the pendulum and quantum oscillator; no new entities or fitted parameters are introduced beyond those already present in the cited physical systems.

axioms (2)
  • standard math Standard back-propagation and optimization algorithms converge to useful minima for the described architectures
    Invoked implicitly when reporting training outcomes for ANN, CNN, LSTM, and PINN models.
  • domain assumption The governing differential equations for the driven damped pendulum and the quantum anharmonic oscillator are correctly transcribed into the loss functions
    Required for the PINN claims to hold; location is the description of the two physical systems.

pith-pipeline@v0.9.0 · 5800 in / 1288 out tokens · 31831 ms · 2026-05-23T03:55:23.490326+00:00 · methodology

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Reference graph

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8 extracted references · 8 canonical work pages · 1 internal anchor

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