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arxiv: 1907.11806 · v1 · pith:XHI4BNXCnew · submitted 2019-07-26 · 💻 cs.LG · math.DS· physics.class-ph· stat.ML

Learning and Interpreting Potentials for Classical Hamiltonian Systems

Harish S. Bhat This is my paper

Pith reviewed 2026-05-24 15:25 UTC · model grok-4.3

classification 💻 cs.LG math.DSphysics.class-phstat.ML
keywords neural networksequation discoveryHamiltonian systemspotential energy functionsinterpretable modelscentral force problemsCoulomb potentialtrajectory data
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The pith

Neural networks trained on trajectories can recover exact algebraic potential functions for classical Hamiltonian systems via equation discovery.

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

The paper shows how to learn a potential energy function for separable classical Hamiltonian systems directly from observed trajectories. A neural network first approximates the potential, after which an equation discovery step converts the network output into a closed-form algebraic expression. Demonstrations on oscillators, central force problems, and charged particles under Coulomb interaction produce close matches to ground truth, including recovery of the correct effective potential in the central force case. A sympathetic reader cares because the method turns opaque learned models into interpretable symbolic equations that can be inspected or used analytically.

Core claim

By constructing a neural network model of the potential and then applying an equation discovery technique, the approach extracts closed-form algebraic expressions from trained networks that agree closely with ground truth potentials, as verified on test problems including oscillators, a central force problem where the correct effective potential is recovered, and two charged particles in a classical Coulomb potential.

What carries the argument

A neural network that models the potential energy, followed by an equation discovery technique that extracts a closed-form algebraic expression from the trained network.

If this is right

  • The method recovers exact potentials for multiple oscillator systems from trajectory data alone.
  • For central force problems the approach yields the correct effective potential as a reduced-order model.
  • The pipeline recovers the Coulomb interaction potential for two charged particles.
  • Close numerical agreement holds between the original neural network, the extracted algebraic form, and the true potential across the tested systems.

Where Pith is reading between the lines

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

  • The same two-step process could be tested on systems where the potential is unknown to generate candidate symbolic models for later physical validation.
  • If equation discovery succeeds on neural potentials, the technique might extend to learning other conserved quantities or force laws from trajectory data.
  • Success on separable Hamiltonians raises the question of whether similar recovery is possible when the Hamiltonian is non-separable.

Load-bearing premise

Observed trajectories contain enough information to uniquely determine the potential, and the equation discovery step can reliably recover the exact algebraic form from the trained neural network.

What would settle it

Apply the full pipeline to a simple harmonic oscillator with known quadratic potential; if the equation discovery step returns a non-quadratic expression despite the neural network accurately reproducing the trajectories, the central claim is false.

Figures

Figures reproduced from arXiv: 1907.11806 by Harish S. Bhat.

Figure 1
Figure 1. Figure 1: For the simple harmonic oscillator (6), after adjusting a constant bias, the neural potential Vb(q) closely matches the true potential V (q) = q 2 /2. with λ = 1 and tune λ downward until the error kk—between the neural network potential Vb(q) and the SINDy-computed approximation—drops below 10−10. We find that with λ = 0.04, the estimated system is Vb(q) ≈ β0 + β2q 2 (7) with β0 ≈ −49.18 and β2 ≈ 0.4978.… view at source ↗
Figure 2
Figure 2. Figure 2: For the double well potential (8), the neural potential Vb(q) trained on T1 closely matches the true potential V (q). This training set includes one high-energy trajectory that visits both wells. In red, we plot V (q) for q ∈ T1; in green, we plot V (q) for q ∈ [−1, 3] \ T1. Potentials were adjusted by a constant bias so that they both have minimum values equal to zero. We seek to understand how the choice… view at source ↗
Figure 3
Figure 3. Figure 3: For the double well potential (8), we train neural potentials Vb(q) using, in turn, the training sets T2 (left) and T3 (right). We plot in red Vb(q) only for the values of q covered by the respective training sets; in green, we extrapolate Vb(q) to values of q that are not in the respective training sets. Since T2 includes two trajectories, one from each well, the neural potential captures and extrapolates… view at source ↗
Figure 4
Figure 4. Figure 4: For the central force problem (10), after adjusting a constant bias, the neural potential Vb(r) closely matches the effective potential Veff(r). 500000 steps at a learning rate of 10−3 . We carry out the training in two stages because training directly with softplus activations and exponential output failed. For this problem, classical physics gives us an effective potential Veff(r) = r −1 + (10 − r) −1 + … view at source ↗
Figure 5
Figure 5. Figure 5: Here we plot both training (left) and test (right) results for the Coulomb problem (14). For both plots, we have subtracted a constant bias, the maximum value of the neural potential on the data set in question. These results are for a neural potential Vb that is a function of the difference q1−q2 between the two charged particles’ positions; for each q in the training and test sets, we plot Vb(q1 − q2) ve… view at source ↗
Figure 6
Figure 6. Figure 6: Here we plot both training (left) and test (right) results for the Coulomb problem (14). For both plots, we have subtracted a constant bias, the maximum value of the neural potential on the data set in question. These results are for a neural potential Vb that is a function of the distance r = kq1 − q2k between the two charged particles; for each q in the training and test sets, we plot Vb(r) versus r = kq… view at source ↗
read the original abstract

We consider the problem of learning an interpretable potential energy function from a Hamiltonian system's trajectories. We address this problem for classical, separable Hamiltonian systems. Our approach first constructs a neural network model of the potential and then applies an equation discovery technique to extract from the neural potential a closed-form algebraic expression. We demonstrate this approach for several systems, including oscillators, a central force problem, and a problem of two charged particles in a classical Coulomb potential. Through these test problems, we show close agreement between learned neural potentials, the interpreted potentials we obtain after training, and the ground truth. In particular, for the central force problem, we show that our approach learns the correct effective potential, a reduced-order model of the system.

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

2 major / 0 minor

Summary. The paper proposes a two-step approach for classical separable Hamiltonian systems: train a neural network to model the potential energy from observed trajectories, then apply an equation discovery technique to extract a closed-form algebraic expression from the trained network. Demonstrations on oscillators, a central-force problem, and two charged particles under Coulomb interaction claim close agreement between the neural potential, the discovered expression, and ground truth, with particular emphasis on recovering the correct effective potential as a reduced-order model in the central-force case.

Significance. If the empirical demonstrations were supported by quantitative metrics and analysis of the recovery step, the method would provide a concrete route from trajectory data to both accurate neural models and interpretable closed-form potentials, with value for reduced-order modeling in Hamiltonian systems.

major comments (2)
  1. [Abstract] Abstract: the central claim of 'close agreement between learned neural potentials, the interpreted potentials we obtain after training, and the ground truth' (including recovery of the effective potential) is asserted without any quantitative metrics, error bars, training details, network architecture, loss functions, or description of the equation discovery algorithm and its basis library.
  2. [Abstract and test-problem demonstrations] The manuscript supplies no analysis or bounds establishing when trajectories determine the potential uniquely (up to irrelevant constants) or when the trained network lies close enough to the true function for the symbolic regressor to return the exact algebraic form rather than an approximation or spurious term; this assumption is load-bearing for the reliability of the interpreted-potential step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We respond to each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of 'close agreement between learned neural potentials, the interpreted potentials we obtain after training, and the ground truth' (including recovery of the effective potential) is asserted without any quantitative metrics, error bars, training details, network architecture, loss functions, or description of the equation discovery algorithm and its basis library.

    Authors: We agree that the abstract would be strengthened by including quantitative support for the claims. In the revised manuscript we will augment the abstract with specific metrics from the experiments (such as mean squared error between the neural-network potential and ground truth, and between the symbolically recovered expression and ground truth) together with concise references to the network architecture, loss function, and equation-discovery procedure (including the basis library) that are already detailed in the methods section. revision: yes

  2. Referee: [Abstract and test-problem demonstrations] The manuscript supplies no analysis or bounds establishing when trajectories determine the potential uniquely (up to irrelevant constants) or when the trained network lies close enough to the true function for the symbolic regressor to return the exact algebraic form rather than an approximation or spurious term; this assumption is load-bearing for the reliability of the interpreted-potential step.

    Authors: The manuscript is primarily empirical and does not contain general theoretical bounds on identifiability or on the accuracy threshold required for exact symbolic recovery. We will add a dedicated discussion subsection that summarizes the empirical conditions (trajectory length, sampling density, and network fitting error) under which exact recovery occurred in the reported test problems and notes the absence of spurious terms in those cases. A full theoretical characterization of uniqueness and recovery guarantees lies beyond the scope of the present work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper describes a two-step procedure: (1) train a neural network to model the potential from observed trajectories of a separable Hamiltonian system, then (2) apply an equation-discovery technique to extract a closed-form expression from the trained network. The abstract and reader's summary provide no equations or self-citations that reduce the claimed recovery of the effective potential (or any other result) to the input data by construction. The central demonstration is empirical agreement on test problems; success is not asserted via a fitted parameter renamed as a prediction, a self-referential definition, or a load-bearing self-citation chain. This is the normal, non-circular case for a data-driven modeling paper.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract identifies the separability assumption for classical Hamiltonians and the use of standard neural-network training plus equation discovery; no other free parameters, axioms, or invented entities are stated.

free parameters (1)
  • neural network parameters
    Weights and biases of the neural network are fitted to trajectory data to approximate the potential.
axioms (1)
  • domain assumption The Hamiltonian is separable into kinetic plus potential terms
    Explicitly stated as the setting for classical separable Hamiltonian systems.

pith-pipeline@v0.9.0 · 5646 in / 1212 out tokens · 29784 ms · 2026-05-24T15:25:56.468681+00:00 · methodology

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Reference graph

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