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Bridging the Physics-Data Gap with FNO-Guided Conditional Flow Matching: Designing Inductive Bias through Hierarchical Physical Constraints

Tsuyoshi Okita

TL;DR

This work addresses the physics-data gap in time-series generation by embedding a hierarchical physics-informed inductive bias into a deep generative framework that unifies operator learning and probabilistic generation. It introduces HPC-FNO-CFM, which combines Fourier Neural Operators to learn physical operators across four hierarchical levels (conservation, dynamics, boundary, empirical) with Conditional Flow Matching and a real-time FNO guidance mechanism. A time-dependent hierarchical constraint loss is used alongside condition-conditioned operator outputs, and theoretical guarantees of well-posedness are provided, demonstrated across harmonic oscillators, human activity recognition, and battery SOH experiments, including an automatic discovery of a Temperature-Capacity Conservation Law. Theoretical results rely on existence/uniqueness of the generative ODE under a Lipschitz velocity field and a bound on FNO error via Grönwall's inequality, supporting the approach's reliability and generalizability.

Abstract

Conventional time-series generation often ignores domain-specific physical constraints, limiting statistical and physical consistency. We propose a hierarchical framework that embeds the inherent hierarchy of physical laws-conservation, dynamics, boundary, and empirical relations-directly into deep generative models, introducing a new paradigm of physics-informed inductive bias. Our method combines Fourier Neural Operators (FNOs) for learning physical operators with Conditional Flow Matching (CFM) for probabilistic generation, integrated via time-dependent hierarchical constraints and FNO-guided corrections. Experiments on harmonic oscillators, human activity recognition, and lithium-ion battery degradation show 16.3% higher generation quality, 46% fewer physics violations, and 18.5% improved predictive accuracy over baselines.

Bridging the Physics-Data Gap with FNO-Guided Conditional Flow Matching: Designing Inductive Bias through Hierarchical Physical Constraints

TL;DR

This work addresses the physics-data gap in time-series generation by embedding a hierarchical physics-informed inductive bias into a deep generative framework that unifies operator learning and probabilistic generation. It introduces HPC-FNO-CFM, which combines Fourier Neural Operators to learn physical operators across four hierarchical levels (conservation, dynamics, boundary, empirical) with Conditional Flow Matching and a real-time FNO guidance mechanism. A time-dependent hierarchical constraint loss is used alongside condition-conditioned operator outputs, and theoretical guarantees of well-posedness are provided, demonstrated across harmonic oscillators, human activity recognition, and battery SOH experiments, including an automatic discovery of a Temperature-Capacity Conservation Law. Theoretical results rely on existence/uniqueness of the generative ODE under a Lipschitz velocity field and a bound on FNO error via Grönwall's inequality, supporting the approach's reliability and generalizability.

Abstract

Conventional time-series generation often ignores domain-specific physical constraints, limiting statistical and physical consistency. We propose a hierarchical framework that embeds the inherent hierarchy of physical laws-conservation, dynamics, boundary, and empirical relations-directly into deep generative models, introducing a new paradigm of physics-informed inductive bias. Our method combines Fourier Neural Operators (FNOs) for learning physical operators with Conditional Flow Matching (CFM) for probabilistic generation, integrated via time-dependent hierarchical constraints and FNO-guided corrections. Experiments on harmonic oscillators, human activity recognition, and lithium-ion battery degradation show 16.3% higher generation quality, 46% fewer physics violations, and 18.5% improved predictive accuracy over baselines.

Paper Structure

This paper contains 32 sections, 2 theorems, 26 equations, 1 figure, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction: Inductive Bias Design in Deep Learning
  2. Related Work: Intersection of Deep Learning and Physics
  3. Neural Operator Theory
  4. Conditional Flow Matching
  5. Physics-Informed Generative Models
  6. Proposed Method: Hierarchical Physics-Constrained Architecture
  7. Overall Design Philosophy
  8. Learning Physical Operators via FNO
  9. Spectral Convolution Layers
  10. Hierarchical Operator Construction
  11. Integrated FNO Output
  12. Probabilistic Generation via CFM
  13. Continuous Normalizing Flow
  14. Condition Encoding
  15. Introduction of FNO Guidance Mechanism
  16. ...and 17 more sections

Key Result

Theorem 1

Assuming that the velocity field integrating Conditional Flow Matching (CFM) with FNO guidance, as used in our proposed method, is Lipschitz continuous and bounded, the ODE governing the generative process admits a unique solution for any initial condition picard1890coddington1955theory.

Figures (1)

  • Figure 1: Overview of architecture of HPC-FNO-CFM.

Theorems & Definitions (6)

  • Theorem 1: Existence and Uniqueness of the Generative ODE Solution
  • proof : Proof Sketch
  • Theorem 2: Boundedness of FNO Approximation Error Impact
  • proof : Proof Sketch
  • proof : Proof of Theorem 1
  • proof : Proof of Theorem 2