Symplectic Neural Flows for Modeling and Discovery

TL;DR

Addressing long-term integration of Hamiltonian systems, the paper introduces SympFlow, a time-dependent symplectic neural flow built from parameterized Hamiltonian flow maps that can either approximate the exact Hamiltonian flow or learn an underlying Hamiltonian from trajectory data. It establishes a universal approximation property for SympFlow in the space of Hamiltonian flows and provides a posteriori backward analysis via Hamiltonian matching, leveraging a long-time extension to a non-autonomous Hamiltonian. The approach models both conservative and non-conservative dynamics by employing a doubled-phase-space formulation for dissipation and demonstrates improved energy conservation and qualitative trajectory fidelity on the simple harmonic oscillator, damped harmonic oscillator, and Henon–Heiles system, with data-efficient supervised learning compared to unconstrained baselines. These results suggest practical impact for reliable, physics-informed learning and long-horizon simulations, with potential extensions to higher-dimensional systems and PDEs and avenues for improving computational efficiency on larger-scale problems.

Abstract

Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose, but neural network-based methods that incorporate these principles remain underexplored. This work introduces SympFlow, a time-dependent symplectic neural network designed using parameterized Hamiltonian flow maps. This design allows for backward error analysis and ensures the preservation of the symplectic structure. SympFlow allows for two key applications: (i) providing a time-continuous symplectic approximation of the exact flow of a Hamiltonian system--purely based on the differential equations it satisfies, and (ii) approximating the flow map of an unknown Hamiltonian system relying on trajectory data. We demonstrate the effectiveness of SympFlow on diverse problems, including chaotic and dissipative systems, showing improved energy conservation compared to general-purpose numerical methods and accurate

Figures (14)

• Figure 1: One layer of the SympFlow
• Figure 2: Training data with $N=100$ initial conditions, $M=50$ sampling times for each of them, and no noise, i.e., $\varepsilon_m^n=0$ for every $n$ and $m$.
• Figure 3: Unsupervised experiment --- Simple Harmonic Oscillator: Comparison of the results obtained with predictions up to time $T=1000$ and $\Delta t=1$.
• Figure 4: Supervised experiment --- Simple Harmonic Oscillator: Comparison of the MLP and the SympFlow trained with a dataset of parameters $N=100$, $M=50$, and $\varepsilon=0$.
• Figure 5: Supervised experiment --- Simple Harmonic Oscillator: Impact of the parameters $N$, $M$, and $\varepsilon$ on the training procedure.

Theorems & Definitions (11)

• Proposition 1: Proposition 1.4.D polterovichGeometryGroupSymplectic2001
• Theorem 1: Universal approximation theorem for SympFlow
• proof : Outline of the proof of \ref{['thm:universal']}
• Remark 1
• Theorem 2: A-posteriori error estimate
• proof : Proof of \ref{['thm:aposteriori']}
• Lemma 1: Approximation with polynomial Hamiltonian
• proof
• Lemma 2: Lemma 1 in turaev2002polynomial
• Corollary 1