Source Anchoring for Physical Consistency in Flow Matching Models

2026-09-27

AI Summary

This study addresses the challenge that existing generative models struggle to ensure samples satisfy physical laws, while post-hoc projection corrections often induce distribution shifts. To overcome this, we propose a source-anchoring approach that encodes physical constraints into the noise source distribution prior to generation, replacing conventional post-processing projections and preventing samples in high-noise regimes from deviating from the target distribution. Building upon this, we develop a functional flow matching framework that integrates source projection with a tailored training objective. Experimental results demonstrate that the proposed method accurately reproduces target distributions across multiple partial differential equation (PDE) tasks, effectively balancing the precision of physical constraints with the fidelity of the generated distribution.

Abstract

Deep generative models are used to solve partial differential equations and model distributions of physical system states, but ensuring that the generated samples satisfy the governing laws remains challenging. Projection-based flow-matching methods enforce physics by correcting the flow from an unconstrained noise distribution. These corrections shift the generated samples away from the distribution of target solutions, especially in high noise regions. To address this limitation, we propose Source Anchoring for Physical Consistency (SAPC), a Functional Flow Matching method that encodes the physical constraints into the source noise before generation begins. We evaluate SAPC on five systems governed by partial differential equations, covering six tasks with linear and non-linear dynamics, and compare results against five baselines and the unconstrained backbone. Anchoring the source reduces the need for large corrections that drive samples onto admissible but off-distribution states, and SAPC reproduces the target distributions most accurately on every evaluated task, while matching the constraint precision of the best projection-based baselines. Ablation experiments show that this gain arises from pairing source projection with a matched training objective that regresses toward the projected source. These results identify the source distribution as a key design choice for physically consistent generative modelling.

Problem

• Flow Matching Models
• Physical Consistency
• Partial Differential Equations
• Deep Generative Models
• Distribution Shift

Innovation

• Flow Matching
• Physical Consistency
• Source Anchoring
• Partial Differential Equations
• Deep Generative Models