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arxiv: 2606.27029 · v1 · pith:XVWO3HWQnew · submitted 2026-06-25 · 💻 cs.LG

Symplectic Neural Networks for learning Generalized Hamiltonians

Harsh Choudhary , Vyacheslav Kungurtsev , Chandan Gupta , Melvin Leok , Georgios Korpas This is my paper

Pith reviewed 2026-06-26 05:30 UTC · model grok-4.3

classification 💻 cs.LG
keywords Hamiltonian Neural Networkssymplectic integratorsimplicit methodsadjoint sensitivitysystem identificationenergy conservationbackward error analysisneural ODEs
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The pith

Symplectic discretizations of the adjoint system match backpropagation sensitivities to enable efficient training of implicit Hamiltonian Neural Networks.

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

The paper aims to establish that symplectic discretizations of the adjoint system produce the same sensitivities as backpropagation, allowing efficient parameter training for Hamiltonian Neural Networks that rely on implicit symplectic integrators. This matters for a sympathetic reader because implicit methods normally preserve energy and geometric structure better for long-term Hamiltonian dynamics but are too slow for training on noisy trajectory data. The authors address the cost with predictor-corrector ODE solvers and fixed point iteration for gradient updates. Experiments on non-separable chaotic systems show advantages in system identification, energy preservation, and reduced memory use, with backward error analysis post-processing yielding a modified Hamiltonian closer to the true one.

Core claim

By leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range

What carries the argument

The equivalence between symplectic discretizations of the adjoint system and backpropagation sensitivities, which carries the argument by allowing gradient computation for implicit integrators without explicit differentiation through the solver.

If this is right

  • Training of implicit symplectic HNNs becomes feasible with predictor-corrector solvers and fixed point iteration for gradient updates.
  • Learned models achieve better system identification and energy preservation on non-separable chaotic systems than standard approaches.
  • Backward error analysis post-processing produces a modified Hamiltonian that approximates the true one more closely without higher-order discretizations.
  • Overall computation and memory complexity for gradient generation is reduced compared to direct backpropagation through implicit steps.

Where Pith is reading between the lines

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

  • The sensitivity-matching property could extend to other structure-preserving integrators used in neural differential equations.
  • Post-processed modified Hamiltonians might allow trading off discretization accuracy against correction steps in related geometric learning settings.
  • The efficiency gains may support scaling the method to higher-dimensional physical systems where implicit methods were previously prohibitive.

Load-bearing premise

Fixed point iteration and predictor-corrector methods can efficiently solve the implicit equations without introducing significant errors that affect the learned Hamiltonian.

What would settle it

A numerical check on a simple known Hamiltonian where the symplectic adjoint gradients differ from standard backpropagation gradients, or where long-term energy drift in the learned model matches that of non-symplectic HNNs.

Figures

Figures reproduced from arXiv: 2606.27029 by Chandan Gupta, Georgios Korpas, Harsh Choudhary, Melvin Leok, Vyacheslav Kungurtsev.

Figure 1
Figure 1. Figure 1: , where φt is the exact flow map for the ODE. It is in this context that the true power of symplectic integrators for simulating Hamiltonian dynamics is revealed. In general, a one-step method cannot be exactly expressed as the time-h flow of a differential equation, but we can construct an asymptotic expansion for this modified equation, y˜˙ = fh(˜y) = f(˜y) + hf1(˜y) + h 2 f2(˜y). . . , y˜(0) = y0 (13) W… view at source ↗
Figure 2
Figure 2. Figure 2: The schematic of the Hamiltonian Identification framework, where the network represents a [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The model architecture of our Neural Network (the number of hidden layers and input dimension is [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative plots for (a) distribution of training data (b) training and validation loss (c) true [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Representative plots for (a) distribution of training data (b) training and validation loss (c) true [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Representative plots for (a) distribution of training data (b) training and validation loss (c) true [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Various orbits in Kepler’s potential (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Representative plots for (a) distribution of training data (b) training and validation loss (c) true [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The Hamiltonian prediction error ∥Htrue − HNN ∥1 in the double well system on test data drawn from 3 different distributions: (a) random uniform; (b) uniform square grid; (c) multivariate Gaussian N (0, I2). (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The Hamiltonian prediction error ∥Htrue − HNN ∥1 in the coupled harmonic oscillator system on test data drawn from 3 different distributions: (a) random uniform; (b) uniform square grid; (c) multivariate Gaussian N (0, I2). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The Hamiltonian prediction error ∥Htrue − HNN ∥1 for the Hénon–Heiles system on test data drawn from 3 different distributions: (a) random uniform; (b) uniform square grid; (c) multivariate Gaussian N (0, I2). (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The Hamiltonian prediction error ∥Htrue − HNN ∥1 in Kepler’s potential system on test data drawn from 3 different distributions: (a) random uniform; (b) uniform square grid; (c) multivariate Gaussian N (0, I2). Here, we only show the two coordinates qx0 , px0 which we sample from the distributions, while the rest of the coordinates are fixed to their respective median values in the test set space. constan… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of (a) memory and (b) runtime profiles for adjoint and backdrop-based gradient [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The Hamiltonian learning process for a second-order symplectic map [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Left The error contours for |Htrue − HNN | and Right |Htrue − H˜NN | where the modified Hamiltonian is written up to fourth-order (contours now show the O(h 4 ) errors). We can see that the modified Hamiltonian of the learned Hamiltonian is a better approximation of the true Hamiltonian than the learned Hamiltonian. In our work, we adopt such an approach, using implicit symplectic partitioned Runge–Kutta … view at source ↗
read the original abstract

Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

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 / 2 minor

Summary. The paper introduces Symplectic Neural Networks (SNNs) for learning generalized Hamiltonians from noisy trajectory observations. It replaces standard explicit integrators in HNNs with implicit symplectic ones, asserts that a symplectic discretization of the adjoint system produces identical parameter sensitivities to back-propagation through the flow map, and employs predictor-corrector schemes plus fixed-point iteration to keep the implicit steps tractable. Experiments on non-separable chaotic systems are reported to show improved long-term energy preservation and system identification; a post-processing step via backward error analysis is claimed to yield a modified Hamiltonian closer to the true one without requiring higher-order discretizations.

Significance. If the central equivalence between the symplectic discrete adjoint and back-propagation holds and the implicit solver residuals remain controlled, the method would make geometrically faithful integrators practical for HNN training, addressing a recognized computational bottleneck. The backward-error-analysis post-processing is a potentially useful, low-cost refinement. No machine-checked proofs or fully reproducible code artifacts are mentioned.

major comments (2)
  1. [Method / Implicit solver description] The central efficiency claim rests on the assertion that a finite number of fixed-point or predictor-corrector iterations produces a trajectory and discrete-adjoint gradient sufficiently close to the exact implicit symplectic step. No quantitative bound relating iteration count, residual norm, and gradient error is supplied, nor is an analysis given of how such residuals propagate under the chaotic dynamics highlighted in the experiments.
  2. [Adjoint discretization section] The statement that 'symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation' is load-bearing for the entire training procedure. The manuscript must exhibit the precise discrete adjoint equations and demonstrate their equivalence to the chain-rule gradients through the implicit map; without this derivation the computational advantage cannot be verified.
minor comments (2)
  1. Notation for the learned Hamiltonian, the implicit map, and the modified Hamiltonian obtained by backward error analysis should be introduced once and used consistently.
  2. The experimental section would benefit from explicit reporting of iteration counts per time step and residual tolerances used in the fixed-point solver across all compared methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and commit to revisions that strengthen the manuscript without altering its core contributions.

read point-by-point responses

  1. Referee: [Method / Implicit solver description] The central efficiency claim rests on the assertion that a finite number of fixed-point or predictor-corrector iterations produces a trajectory and discrete-adjoint gradient sufficiently close to the exact implicit symplectic step. No quantitative bound relating iteration count, residual norm, and gradient error is supplied, nor is an analysis given of how such residuals propagate under the chaotic dynamics highlighted in the experiments.

    Authors: We agree that a quantitative analysis of residual propagation would improve rigor. In the revision we will add a dedicated subsection on the fixed-point iteration, deriving a contraction-mapping bound on the residual after a fixed number of iterations and relating it to the gradient error via the implicit function theorem. We will also augment the experiments with a sensitivity study that varies iteration count on the chaotic test systems and reports the resulting variation in learned Hamiltonian accuracy and long-term energy drift. revision: yes

  2. Referee: [Adjoint discretization section] The statement that 'symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation' is load-bearing for the entire training procedure. The manuscript must exhibit the precise discrete adjoint equations and demonstrate their equivalence to the chain-rule gradients through the implicit map; without this derivation the computational advantage cannot be verified.

    Authors: We will expand the Adjoint discretization section to include the full set of discrete adjoint equations for the chosen implicit symplectic scheme together with a self-contained derivation that shows step-by-step equivalence to the chain-rule gradients obtained by differentiating through the implicit map. This addition will make the claimed computational advantage directly verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation leverages established symplectic adjoint equivalence

full rationale

The paper's central claim rests on the known mathematical property that symplectic discretizations of the adjoint system produce the same sensitivities as backpropagation through the flow map. This is invoked as an external fact to enable efficient training, not derived or fitted within the paper itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described method. The predictor-corrector and fixed-point approximations are presented as computational mitigations rather than part of a closed derivation loop. The overall approach is self-contained against external benchmarks on symplectic integrators and adjoint methods.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard assumption that the system obeys Hamiltonian mechanics and that symplectic integrators preserve the necessary structure.

axioms (1)
  • domain assumption The underlying dynamical system is Hamiltonian.
    The entire approach is based on learning the Hamiltonian of the system.

pith-pipeline@v0.9.1-grok · 5793 in / 1145 out tokens · 32634 ms · 2026-06-26T05:30:27.804769+00:00 · methodology

discussion (0)

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Reference graph

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