cs.LGMar 2, 2026

CHLU: The Causal Hamiltonian Learning Unit as a Symplectic Primitive for Deep Learning

Authors: Pratik Jawahar, Maurizio Pierini

Organizations: University of Manchester · CERN

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Abstract

Current deep learning primitives dealing with temporal dynamics suffer from a fundamental dichotomy: they are either discrete and unstable (LSTMs) \citep{pascanu_difficulty_2013}, leading to exploding or vanishing gradients; or they are continuous and dissipative (Neural ODEs) \citep{dupont_augmented_2019}, which destroy information over time to ensure stability. We propose the Causal Hamiltonian Learning Unit (pronounced: clue), a novel Physics-grounded computational learning primitive. By enforcing a Relativistic Hamiltonian structure and utilizing symplectic integration, a CHLU strictly conserves phase-space volume, as an attempt to solve the memory-stability trade-off. We show that the CHLU is designed for infinite-horizon stability, as well as controllable noise filtering. We then demonstrate a CHLU's generative ability using the MNIST dataset as a proof-of-principle.

Figures & tables

Figure 1: Lemniscate tracing experiment. The transparent lines show inference cycles 0−480-48 and the solid lines show the final 22 cycles, for LSTM, NODE and CHLU (left to right)
Figure 2: The LSTM, NODE and CHLU (left to right), each predicts the state qq given the same initial perturbed states, from which we calculate the KE and overlay it on the expected KE.
Figure 3: MNIST digits generated from the centroid of the test set with added noise.
Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: The learned lemniscate potential energy plotted as 2D2D , 3D3D heatmaps (top left, right) as well as a force field (bottom). The output trajectory is overlayed on all 33 for visual comparison.
Figure 5: Kinetic energies calculated from the output state qq (top 3 rows), compared to the kinetic energies calculated from output state pp (bottom 3 rows) for the same 33 sine waves with the same inital perturbed conditions for all models (left to right: LSTM, NODE, CHLU).
Figure 6: Phase space plots corresponding to the kinetic energy plots in Figs. 2 , 5 , with the blue triangles representing the perturbed initial states and the grey ellipse showing the expected trajectory (left to right: LSTM, NODE, CHLU).
Figure 7: MNIST evolution from step 200−1000200-1000 from left to right, at intervals of 200200 steps, for 55 digits (rows).
Figure 8: MNIST evolution from step 200−1000200-1000 from left to right, at intervals of 200200 steps, for 55 digits (rows), in the non-thermal conservative mode.
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