arxiv: 2604.23606 · v1 · submitted 2026-04-26 · 💻 cs.LG · math-ph· math.MP

Recognition: unknown

Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data

Ru Geng , Panayotis Kevrekidis , Yixian Gao , Hong-Kun Zhang , Jian Zu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:23 UTC · model grok-4.3

classification 💻 cs.LG math-phmath.MP

keywords Hamiltonian graph inferencestructure discoverydynamics predictionlattice systemstrajectory dataenergy conservationphysics-informed GNNnon-separable Hamiltonians

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The pith

Hamiltonian Graph Inference Networks jointly recover unknown interaction graphs and predict long-time trajectories from data for lattice Hamiltonian systems.

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

Lattice Hamiltonian systems model many physical phenomena but are difficult to learn from data when both the interaction graph and node uniformity are unknown. The paper introduces HGIN to address this by training a weighted adjacency matrix under a loss derived from Hamilton's equations while clustering edges into subgraphs that receive separate encoders. This joint structure-discovery and prediction setup handles separable and non-separable Hamiltonians as well as heterogeneous nodes. On three lattice benchmarks it produces dramatically lower errors in long-term energy and trajectory forecasts than prior methods. The learned weights also carry information about the parity of the underlying pair potential through a symmetry property of the loss.

Core claim

HGIN couples a structure-learning module, consisting of a learnable weighted adjacency matrix trained under a Hamilton's-equations loss, with a trajectory-prediction module that partitions edges into physically distinct subgraphs via k-means clustering and assigns each subgraph its own encoder. On a Klein-Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (one homogeneous, one heterogeneous), this yields reductions in long-time energy and trajectory prediction error of six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss shows that the learned weights encode the parity of the pair potential.

What carries the argument

Learnable weighted adjacency matrix trained under Hamilton's equations loss, combined with k-means clustering of edges into subgraphs each assigned a dedicated encoder.

If this is right

Joint recovery of graph structure and dynamics becomes possible from trajectory data alone without assuming separability or node homogeneity.
Long-time prediction accuracy improves by many orders of magnitude on standard lattice benchmarks compared with graph-based baselines.
The parity of the pair potential can be read out directly from the learned edge weights via the symmetry of the Hamiltonian loss.
Parameter-sharing limitations of standard GNNs are broken by assigning separate encoders to physically clustered subgraphs.

Where Pith is reading between the lines

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

The same joint-inference strategy could be applied to other physical domains where a known conservation law can serve as the training loss.
Clustering edges by learned weights might reveal multi-range or multi-body interaction scales in more complex lattices or networks.
Extending the method to noisy or partially observed trajectories would test its usefulness on real experimental data from optics or biophysics.

Load-bearing premise

Enforcing Hamilton's equations as a training loss on a learnable adjacency matrix recovers the true interaction structure for non-separable Hamiltonians and heterogeneous nodes without the clustering step introducing artifacts that harm long-term stability.

What would settle it

Training HGIN on a new synthetic lattice whose non-separable Hamiltonian and heterogeneous node parameters are known in advance, then checking whether the recovered adjacency matrix matches the true interactions and whether energy and trajectories remain accurate over thousands of time steps.

Figures

Figures reproduced from arXiv: 2604.23606 by Hong-Kun Zhang, Jian Zu, Panayotis Kevrekidis, Ru Geng, Yixian Gao.

**Figure 1.** Figure 1: FIG. 1. Architecture of the Hamiltonian Graph Inference Net view at source ↗

**Figure 2.** Figure 2: FIG. 2. KG-LRI structure recovery. (A) Weighted adjacency m view at source ↗

**Figure 3.** Figure 3: FIG. 3. Average MSE of predicted trajectories over time on 30 view at source ↗

**Figure 4.** Figure 4: FIG. 4. Learned weighted adjacency matrices and their view at source ↗

**Figure 5.** Figure 5: FIG. 5. Comparing the importance of view at source ↗

**Figure 6.** Figure 6: FIG. 6. Comparing the importance of learning approach for th view at source ↗

**Figure 7.** Figure 7: FIG. 7. Comparison of structural learning methods. A. groun view at source ↗

**Figure 8.** Figure 8: FIG. 8. Noise experiment on KG-LRI benchmark. The three subfi view at source ↗

read the original abstract

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in $\alpha$-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via $k$-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schr\"odinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

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.

Desk Editor's Note private letter to a colleague

HGIN pairs a Hamilton-equation loss on a learnable adjacency matrix with k-means edge clustering to handle non-separable heterogeneous lattices, but the six-to-thirteen-order error drops need tighter experimental controls.

read the letter

The core move here is training a weighted adjacency matrix directly under Hamilton's equations, then clustering the resulting edges once with k-means so each cluster gets its own encoder. That breaks the uniform parameter sharing that limited earlier graph Hamiltonian networks to separable homogeneous cases. The symmetry argument that follows from the loss is clean: the learned weights end up reflecting the parity of the pair potential without any extra supervision. On the three lattice benchmarks the reported long-time energy and trajectory errors drop sharply compared with the baselines they chose. If those numbers hold under scrutiny, the method gives a practical route to joint structure discovery and rollout prediction when you have only state trajectories and no prior graph or homogeneity assumption. The experimental section is the soft spot. The abstract gives no concrete information on how the baselines were re-implemented, what data splits were used, or how sensitive the k-means step is to initialization and choice of k. Without those checks it is hard to know whether the clustering isolates genuine interaction types or simply reflects fitting artifacts that happen to stabilize the particular test lattices. The stress-test worry about downstream stability therefore lands until the authors show ablations on the clustering and confirm that the symmetry property survives outside the training loss. This is aimed at people who model lattice systems in condensed-matter or nonlinear optics and want to infer topology from data rather than assume it. The idea is coherent and the claims are falsifiable, so it deserves a serious referee even though the current evidence is still preliminary. I would send it to review and ask specifically for the missing baseline details and clustering robustness tests.

Referee Report

3 major / 2 minor

Summary. The paper introduces the Hamiltonian Graph Inference Network (HGIN) for jointly recovering the unknown interaction graph and predicting long-time trajectories in lattice Hamiltonian systems from state data. It couples a learnable weighted adjacency matrix trained solely under a Hamilton's-equations loss with k-means clustering of edges to create separate encoders per subgraph, thereby addressing both separable/non-separable Hamiltonians and homogeneous/heterogeneous node dynamics. On three benchmarks (Klein-Gordon lattice with long-range interactions and homogeneous/heterogeneous discrete nonlinear Schrödinger lattices), HGIN is reported to reduce long-time energy and trajectory prediction errors by six to thirteen orders of magnitude relative to baselines, while a symmetry argument on the Hamiltonian loss shows that the learned edge weights encode the parity of the underlying pair potential.

Significance. If the performance claims prove robust, the work would be significant for data-driven modeling of physical lattice systems where both topology and node homogeneity are unknown a priori. The non-circular symmetry argument provides a useful interpretability tool, and the joint structure-discovery plus dynamics-prediction formulation extends beyond prior graph Hamiltonian networks that assume separability or given graphs.

major comments (3)

[Abstract and Experiments] Abstract and Experiments section: the central claim of 6–13 order-of-magnitude reductions in long-time energy and trajectory errors lacks supporting details on baseline implementations (e.g., how the α-separable graph Hamiltonian network and other GNN baselines were trained and tuned), data splits, hyperparameter sensitivity, or statistical significance across multiple random seeds. Without these, the evidential support for the performance advantage remains limited.
[Method (structure-learning and clustering)] Method section on structure-learning module and clustering: the k-means partitioning of the converged weighted adjacency matrix is performed once in an unsupervised manner after training. For the heterogeneous discrete nonlinear Schrödinger benchmark, it is unclear whether this step correctly isolates physically distinct interaction types or injects partitioning artifacts that affect the parameter-sharing bottleneck and long-term rollout stability; the symmetry argument addresses only parity encoding and does not speak to clustering correctness.
[Symmetry argument] Symmetry argument (derived from the Hamiltonian loss): while non-circular, the manuscript should explicitly verify that the parity-encoding property of the learned weights survives the downstream clustering step and separate encoders when the Hamiltonian is non-separable.

minor comments (2)

[Notation and Methods] Clarify initialization, regularization, and convergence criteria for the learnable adjacency matrix in the methods section.
[Figures] Trajectory and energy plots should explicitly mark the training horizon versus long-time extrapolation to substantiate the stability claims.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us improve the clarity and evidential support of the manuscript. We address each major comment below and have incorporated revisions to strengthen the presentation of results, methodological details, and interpretability arguments.

read point-by-point responses

Referee: [Abstract and Experiments] Abstract and Experiments section: the central claim of 6–13 order-of-magnitude reductions in long-time energy and trajectory errors lacks supporting details on baseline implementations (e.g., how the α-separable graph Hamiltonian network and other GNN baselines were trained and tuned), data splits, hyperparameter sensitivity, or statistical significance across multiple random seeds. Without these, the evidential support for the performance advantage remains limited.

Authors: We agree that additional implementation and statistical details are needed to robustly support the performance claims. In the revised manuscript, we have substantially expanded the Experiments section and added a dedicated 'Implementation Details' subsection. This includes: full specifications of baseline architectures and training procedures (including optimizer, learning rates, and early stopping criteria); hyperparameter tuning methodology with the ranges explored; data split protocols (temporal train/validation/test divisions with ratios); and all quantitative results now reported as means ± standard deviations over five independent random seeds. Hyperparameter sensitivity analyses and ablation tables have been moved to the supplementary material. These changes provide the requested context and establish statistical significance for the reported error reductions. revision: yes
Referee: [Method (structure-learning and clustering)] Method section on structure-learning module and clustering: the k-means partitioning of the converged weighted adjacency matrix is performed once in an unsupervised manner after training. For the heterogeneous discrete nonlinear Schrödinger benchmark, it is unclear whether this step correctly isolates physically distinct interaction types or injects partitioning artifacts that affect the parameter-sharing bottleneck and long-term rollout stability; the symmetry argument addresses only parity encoding and does not speak to clustering correctness.

Authors: We acknowledge the concern about potential artifacts from the post-training k-means step in the heterogeneous case. While the original experiments showed that the partitions aligned with distinct interaction types and yielded stable rollouts, we have added targeted analyses in the revised manuscript to address this directly. Specifically, we include: (1) robustness checks of k-means to random initializations and choice of k, with quantitative alignment metrics (e.g., adjusted Rand index) against ground-truth interaction groupings; (2) ablations comparing rollout stability and error when using the learned clusters versus random or incorrect partitions; and (3) visualizations confirming that clusters separate physically meaningful subgraphs without degrading the parameter-sharing benefits. These additions demonstrate that clustering isolates distinct types without introducing detrimental artifacts, while the symmetry argument remains focused on parity as noted. revision: yes
Referee: [Symmetry argument] Symmetry argument (derived from the Hamiltonian loss): while non-circular, the manuscript should explicitly verify that the parity-encoding property of the learned weights survives the downstream clustering step and separate encoders when the Hamiltonian is non-separable.

Authors: We appreciate this request for explicit verification in the non-separable setting. The symmetry argument follows directly from the structure of the Hamiltonian loss and applies to the learned weights independently of subsequent processing. In the revised manuscript, we have added a dedicated paragraph and supplementary proof showing that the parity-encoding property is preserved after clustering and per-subgraph encoders: because the loss is computed on the full predicted trajectories (which use the clustered encoders), the symmetry constraint on the weights remains enforced. We provide both a formal argument and empirical confirmation on the non-separable benchmark, confirming that the interpretability readout holds post-clustering. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical results and symmetry argument are independent of inputs

full rationale

The paper's core contributions are an architecture (learnable adjacency + Hamilton loss + k-means partitioning into per-subgraph encoders) and benchmark evaluations showing large error reductions on specific lattices. The symmetry argument is presented as a mathematical consequence of the loss form rather than a fitted output renamed as prediction. No derivation step equates a claimed result to its training inputs by construction, no self-citation chain bears the central claim, and no uniqueness theorem is invoked to force the method. Performance numbers are data-driven and falsifiable on held-out rollouts; the clustering step is an unsupervised post-processing choice whose validity is assessed empirically rather than derived tautologically.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach rests on standard Hamiltonian mechanics and graph representation assumptions plus learned parameters; no new physical entities are postulated.

free parameters (2)

learnable weighted adjacency matrix
Trained end-to-end under the Hamilton-equations loss to infer interaction topology.
k in k-means clustering of edges
Chosen to partition edges into physically distinct subgraphs for separate encoders.

axioms (2)

domain assumption The observed dynamics obey Hamilton's equations
Used as the training loss for both structure learning and trajectory prediction.
domain assumption Interactions can be represented by a single weighted adjacency matrix
Core modeling choice for the structure-learning module.

pith-pipeline@v0.9.0 · 5555 in / 1452 out tokens · 31714 ms · 2026-05-08T06:23:47.016666+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

63 extracted references · 7 canonical work pages · 2 internal anchors

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Stan- dard GNN methods require prior knowledge of the graph structure; our approach does not

Our model learns the potential interactions among system particles under the assumption that only trajectory data is available. Stan- dard GNN methods require prior knowledge of the graph structure; our approach does not
[2]

3) Our model can identify whether the system exhibits heterogeneous node dynamics and perform clas- siﬁcation accordingly

Our model applies to both separable and non-separable Hamiltonian systems. 3) Our model can identify whether the system exhibits heterogeneous node dynamics and perform clas- siﬁcation accordingly. The remainder of the paper is organized as follows. Sec. II introduces the mathematical framework. Sec. III describes HGIN. Sec. IV reports numerical results, ...
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Subgraph classiﬁcation The structural learning module outputs a weighted adjacency matrix Wθ = (wij), where each entry wij ∈ R quantiﬁes the interaction strength between particles α i and α j. The weighted adjacency matrix has the following empirically observed property: Property 1 If the interaction potential be- tween particles α i and α j satisﬁes Vij(...
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Per-subgraph Hamiltonian Reusing the node and edge features ( 1) and ( 2), we assign each oﬀ-diagonal cluster Co k its own edge encoder gk enc and each diagonal cluster Cd k its own node encoder gk noc, writing H(t) 1 = v∑ k=1 ∑ (i,j )∈C o k wijgk enc(e(t) ij ), (6) H(t) 2 = u∑ k=1 ∑ i∈C d k gk noc(n(t) i ), (7) H(t) = H(t) 1 + H(t) 2 . (8) Distinct encod...
[5]

Trajectory prediction Once trained, the learned Hamiltonian de- ﬁnes a vector ﬁeld through Hamilton’s equa- tions; integrating this ﬁeld from any initial con- dition ( q0, p0) yields the predicted trajectory (ˆqt, ˆpt) = ( q0, p0) + ∫ t t0 ( ∂H ∂p, − ∂H ∂q ) ds. (10) IV. COMPUTATIONAL RESULTS We evaluate HGIN on three lattice Hamilto- nian benchmarks span...
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10, matching the coeﬃcients of the three coupling terms in Eq

15 : 0. 10, matching the coeﬃcients of the three coupling terms in Eq. ( 11). The diagonal entries is classiﬁed into one class and absorb the on- site kinetic and potential terms 1 2p2 + 1 2q2 + 1 4q4 into a single node-encoder cluster. Thus the four non-noise clusters reproduce the four dis- tinct functional forms of the true Hamiltonian without any prio...
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The weights be- tween particle pairs (α 5,α 10) and (α 6,α 12) are 10 FIG

where 2(pnpn+1 + qnqn+1 +pnpn+3 +qnqn+3)). The weights be- tween particle pairs (α 5,α 10) and (α 6,α 12) are 10 FIG. 3. Average MSE of predicted trajectories over time on 30 test samples (y-axis on a logarithmic scale). Panel A: KG-LRI (Sec. IV A). Panel B: Heterogenous DNLS (Sec. IV C). Panels C and D: homogeneous DNLS with N=4 and N=32 (Sec. IV B). In ...

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Replacement of the graph learning loss In this subsection, we test the importance of the LGL setting. Fig. 5 presents a compari- son of the system’s weighted adjacency matrices learned under diﬀerent LGL settings. Clearly, only our method learns weights that are consis- tent with the ground truth
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Comparison of weighted adjacency matrix module conﬁgurations In this subsection, we test the importance of the adjacency matrix weight learning setup. We replace the learning approach for the weights wij in our model with Graph Attention Neural Network (GAT) [ 49], Neural Relational Infer- ence (NRI) [ 50], and Transformer architecture [51], respectively,...
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The learned weighted adjacency ma- trices are shown in Fig

Comparison of structural learning methods In this section, we compare our structure learning method with those of Equivariant Multi-agent Motion Prediction (EqMotion) [ 52] and NRI. The learned weighted adjacency ma- trices are shown in Fig
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Appendix D: The importance of subgraph learning To isolate the contribution of the subgraph learning strategy, we train an ablation variant which we call HGIN-oracle

Clearly, only our method correctly captures the interactions and their intensity relationships within the system. Appendix D: The importance of subgraph learning To isolate the contribution of the subgraph learning strategy, we train an ablation variant which we call HGIN-oracle. In HGIN-oracle, the weighted adjacency matrix Wθ is ﬁxed. To be more precise...
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