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CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics

The paper introduces CoSynFlow, a structure-preserving neural flow that combines symplectic shear maps with explicit conformal scaling to accurately predict long-term dynamics of dissipative Hamiltonian systems across unseen configurations without retraining.

Original authors: Baige Xu, Takaharu Yaguchi

Published 2026-08-04

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CC BY 4.0

Original authors: Baige Xu, Takaharu Yaguchi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future path of a swinging pendulum or a planet orbiting a star. In the world of physics, these movements aren't just random squiggles; they follow strict, invisible rules that keep the universe balanced. Scientists call these "Hamiltonian dynamics." Think of them like a perfectly choreographed dance where energy is never lost, just swapped between speed and position. For decades, computers have been great at learning these dances for simple, perfect systems. But the real world is messy. Things like air resistance or friction act like a "dissipative" force, slowly stealing energy from the system and making the dance slow down and shrink.

The tricky part is that when energy is lost, the mathematical rules change. Instead of keeping the dance floor's volume exactly the same, the floor itself starts to shrink at a specific rate. Most computer models designed to predict these movements either ignore this shrinking (pretending the floor is still the same size) or try to guess the shrinking rate by trial and error, which often leads to the model getting confused and the predictions falling apart over time. Researchers want a way to teach computers to respect these shrinking rules automatically, so they can predict how complex, energy-losing systems behave not just for one specific setup, but for any setup without having to relearn everything from scratch.

This is where the new paper, CoSynFlow, steps in with a clever solution. The authors, Baige Xu and Takaharu Yaguchi, have built a special kind of "neural flow"—a smart computer program designed to mimic the movement of these dissipative systems. You can think of their model as a master choreographer who doesn't just memorize one dance routine but learns the grammar of how energy loss works.

Here is how it works: The model is built like a series of building blocks. Some blocks handle the "shearing" motion (the standard dance moves), while other blocks are explicitly programmed to "shrink" the dance floor at the exact rate the physics demands. The magic happens because the model is "conditioned" on the system's details. Imagine you give the model a description of the specific pendulum you are watching (its weight, length, and how much friction it has). The model then instantly adjusts its internal gears to match that specific system.

The paper proves that this approach is incredibly robust. Because the shrinking rule is hard-coded into the model's architecture, it never makes a mistake about how the system should shrink; it keeps the "structure error" at the level of a computer's tiniest possible rounding error (machine precision). This means that even if you ask the model to predict the movement far into the future—longer than it was ever trained to see—it doesn't drift off course. In tests, CoSynFlow predicted the paths of four different complex systems (like a coupled Duffing system and a Mexican hat potential) with much higher accuracy than other methods. While other models would eventually spiral out of control or settle in the wrong spot, CoSynFlow kept the trajectory tight and true, even when the dissipation (friction) was stronger than anything it had seen during training.

Perhaps most impressively, the model learned to predict the behavior of systems it had never seen before without needing any new training data. It's like teaching a musician to play a song in a new key just by showing them the sheet music once, rather than making them practice the new song for hours. The authors also showed that by adding a "physics-informed" check—where the model is penalized if its movement doesn't match the underlying math equations—it could learn even faster and more accurately, especially when there wasn't much data available.

In short, CoSynFlow is a new tool that forces artificial intelligence to respect the fundamental laws of energy loss. It doesn't just guess; it builds the rules of the universe directly into its brain, allowing it to predict the long-term fate of complex, energy-draining systems with a level of precision and flexibility that previous methods simply couldn't match.

Technical Summary: CoSynFlow – Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics

Problem Statement
Learning solution operators for differential equations is a central objective in scientific machine learning. While neural operators can represent solutions across varying initial conditions and system parameters, they typically treat the solution map as an unconstrained function approximation, failing to enforce the geometric structure of the underlying dynamics. Existing structure-preserving methods, such as SympNets and Symplectic Neural Flows, are designed for conservative Hamiltonian systems where the symplectic form is strictly preserved. However, many physical systems are dissipative Hamiltonian systems, where the symplectic form evolves according to a conformal factor determined by dissipation ( $\phi_t^* \omega = e^{\gamma t}\omega$ ).

Current approaches face three specific challenges in this setting:

Incorrect Constraints: Ordinary symplectic preservation is inappropriate for dissipative systems; the model must preserve the conformal symplectic structure.
Discrete vs. Continuous: Conventional integrators and learned fixed-step maps reach a target time through repeated local updates, lacking a single model that represents solution maps over a continuous range of query times.
Lack of Generalization: Structure-preserving models trained for one Hamiltonian do not generalize to others, requiring separate training for each system in a family.

Methodology: CoSynFlow
The authors propose CoSynFlow, a system-conditioned conformal symplectic neural flow designed to learn continuous-time solution maps for dissipative Hamiltonian dynamics.

Architecture: The model constructs the time- $t$ $t$ map as a composition of symmetric blocks. Each block consists of gradient shear maps (generated by scalar potentials) and explicit conformal scaling layers.
- Shears: The shear maps $P$ and $Q$ are symplectic by construction because they are generated by scalar potentials.
- Scaling: Explicit scaling layers $C$ apply a factor $e^{\gamma t \alpha_\ell}$ to the momentum variables. The scaling factors $\alpha_\ell$ are normalized such that $\sum \alpha_\ell = 1$ .
- Composition: The composition of these blocks ensures the total map satisfies the exact conformal symplectic condition $(\hat{\Phi}_t^\theta)^* \omega = e^{\gamma t} \omega$ for any trainable parameters.
System Conditioning: To enable cross-system prediction, a learned operator (a Fourier Neural Operator) maps a finite-dimensional Hamiltonian descriptor (values of the Hamiltonian at fixed sensor points) and the dissipation parameter $\gamma$ to the block parameters (potentials and scaling allocations). This allows a single trained model to predict solution maps for unseen systems without retraining.
Continuous-Time Input: The query time $t$ is provided as a continuous input to every block, allowing the model to evaluate the solution map at any $t$ and compose maps to predict beyond the training horizon.
Physics-Informed Training: Because the flow is differentiable with respect to time, the residual of the governing differential equation can be evaluated directly, enabling physics-informed training without additional data costs.

Key Contributions

Exact Conformal Symplecticity: By construction, the learned time- $t$ map is exactly conformal symplectic for any trainable parameters, matching the geometric relation of dissipative Hamiltonian dynamics.
Continuous-Time Flow Learning: The model produces the time- $t$ map for any $t$ in a single evaluation. It can be composed to predict beyond the training horizon with the exact conformal factor, and its differentiability in $t$ admits physics-informed training on the governing equation residual.
Cross-System Prediction: A single shared model predicts solution maps for unseen dissipative Hamiltonian systems without retraining, while conditioning leaves the exact geometric structure intact.

Experimental Results
The authors evaluated CoSynFlow on randomly generated Hamiltonian systems and four analytic benchmark systems (anisotropic oscillator, coupled Duffing, Mexican hat potential, quartic coupled oscillator) outside the training distribution.

Accuracy and Structure: CoSynFlow achieved the lowest in-distribution Mean Squared Error (MSE) and maintained the conformal symplectic structure error at machine precision ( $\sim 10^{-15}$ ), significantly outperforming unstructured models, softly constrained models, and standard neural operators.
Long-Horizon Prediction: When predicting over ten times the training horizon by composing learned maps, CoSynFlow attained the lowest state and energy errors. In contrast, models enforcing ordinary symplecticity failed to contract phase-space volume, leading to qualitative errors (e.g., failing to settle into potential wells).
Extrapolation: CoSynFlow demonstrated robustness in extrapolating to dissipation parameters ( $\gamma$ ) up to three times larger than those seen during training, outperforming baselines by a factor of 10 to 20.
Physics-Informed Training: Adding the governing equation residual to the training objective reduced state error by a factor of four and energy error by a factor of six on a single system with limited data.

Significance
The paper claims that CoSynFlow successfully bridges the gap between geometric deep learning and dissipative dynamics. By enforcing the conformal symplectic structure exactly rather than approximately, the model avoids the accumulation of geometric errors that plague standard neural operators and unstructured flows. The ability to generalize across a family of systems without retraining, combined with continuous-time evaluation and exact structure preservation, offers a robust framework for learning solution operators in dissipative physical systems. The work demonstrates that exact geometric constraints do not limit approximation capability (proven via a uniform approximation theorem) and are essential for long-horizon stability in dissipative settings.

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