Least-Action-Guided Diffusion for Physical Extrapolation

2026-06-09

AI Summary

This work addresses the physical inconsistency exhibited by generative models when extrapolating in time, parameters, or geometry by introducing a Least Action Principle–Guided (LAPG) diffusion framework. The method integrates conditional score-based diffusion with physics-informed guidance derived from the action functional during inference: it first generates in-distribution initial states and then refines them toward out-of-distribution target conditions via a variational prior. Crucially, the principle of least action is innovatively reformulated into a differentiable inference correction mechanism, circumventing the need for hard constraints or heuristic loss balancing during training. Evaluated on tasks including free-fall motion, spring-mass oscillators, point-vortex dynamics, and airfoil potential flow, LAPG effectively suppresses phase drift, preserves dissipative properties, accurately captures vortex interactions, and enhances lift prediction accuracy.

Abstract

Reliable extrapolation remains a central challenge for generative models in computational physics, because models trained over finite ranges of time, parameters, or geometries may produce physically inconsistent predictions outside the training distribution. We introduce a least-action-principle-guided diffusion, LAPG, a framework that promotes physical consistency during inference rather than relying solely on constraints imposed during training. The method combines a conditional score-based diffusion model with an action-derived physical guidance score. In the first stage, the learned score model generates an in-distribution proposal; in the second, an action-based variational prior refines this proposal toward the target out-of-distribution condition. This formulation turns the principle of least action into a differentiable inference-time correction mechanism and provides an alternative to pointwise residual penalties that often require empirical loss balancing. We evaluate LAPG on representative ordinary- and partial-differential-equation systems, including free fall, conservative and dissipative spring-mass dynamics, interacting point vortices, and potential flow over parameterized airfoils. In temporal, parameter, and geometric extrapolation tests, LAPG reduces phase drift, preserves dissipative decay, captures vortex motion, and improves the lift response of airfoil flows compared with training-time physics-informed baselines.

Problem

• extrapolation
• physical consistency
• generative models
• computational physics
• out-of-distribution prediction

Innovation

• least-action principle
• diffusion model
• physical extrapolation
• score-based generative model
• variational inference