Source Anchoring for Physical Consistency in Flow Matching Models
cs.LG, cs.AI
Submitted: 2026-09-27
Updated: 2026-09-27
Comments: Submitted to ICLR 2027
License: http://creativecommons.org/licenses/by-nc-sa/4.0/
The gist: 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
Terminology
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.
Sources
- Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling
- Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators
- Divergence-Free Diffusion Models for Incompressible Fluid Flows
- DiffWave: A Versatile Diffusion Model for Audio Synthesis
- Physics-Constrained Diffusion Model for Synthesis of 3D Turbulent Data
- Generative diffusion learning for parametric partial differential equations
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