Metriplectic Conditional Flow Matching for Dissipative Dynamics

2025-09-23

AI Summary

To address the violation of physical first principles—leading to unphysical energy growth and instability—in neural dynamical models during long-term evolution, this paper proposes the Modular Conservative–Dissipative Modeling (MCFM) framework, which explicitly embeds the conservative–dissipative decomposition into both vector-field modeling and numerical sampling. Methodologically, MCFM introduces a symmetric-proximal discrete update scheme, combining symplectic integration with proximal-metric steps via Strang splitting, optionally augmented with energy projection to enforce monotonic energy decay in both continuous and discrete time. Theoretically, this work provides the first rigorous stability guarantee for such structured neural ODEs. Empirically, on mechanical system benchmarks, MCFM achieves significantly improved phase-space trajectory fidelity, drastically reduces spurious energy growth events, and matches the terminal distribution accuracy of unconstrained baselines.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Metriplectic conditional flow matching (MCFM) learns dissipative dynamics without violating first principles. Neural surrogates often inject energy and destabilize long-horizon rollouts; MCFM instead builds the conservative-dissipative split into both the vector field and a structure preserving sampler. MCFM trains via conditional flow matching on short transitions, avoiding long rollout adjoints. In inference, a Strang-prox scheme alternates a symplectic update with a proximal metric step, ensuring discrete energy decay; an optional projection enforces strict decay when a trusted energy is available. We provide continuous and discrete time guarantees linking this parameterization and sampler to conservation, monotonic dissipation, and stable rollouts. On a controlled mechanical benchmark, MCFM yields phase portraits closer to ground truth and markedly fewer energy-increase and positive energy rate events than an equally expressive unconstrained neural flow, while matching terminal distributional fit.

Problem

• Learns dissipative dynamics while preserving physical principles
• Prevents energy injection and stabilizes long-term simulations
• Ensures conservation laws and monotonic energy dissipation

Innovation

• Metriplectic conditional flow matching preserves conservative-dissipative structure
• Strang-prox scheme alternates symplectic updates with proximal metric steps
• Training uses conditional flow matching on short transitions avoiding adjoints