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Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method

Diego Marcondes

TL;DR

This work addresses the statistical properties of physics-informed estimators when the physics is encoded via a linear differential operator. It develops complexity-dependent error bounds using the small-ball method for convex function classes, comparing soft physics-informed penalties to hard constraints and quantifying the statistical gain from incorporating physical knowledge. The key findings show that, under suitable assumptions, the error rates of physics-informed estimators are close to those of hard-constrained empirical minimizers up to constants, and that informed penalization can effectively reduce model complexity by restricting to the kernel of the operator, akin to dimensionality reduction. The results provide a general theoretical framework for evaluating PISL methods beyond specific PDEs, with implications for PINNs and broader physics-informed learning in convex settings.

Abstract

Physics-informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties is still missing. Specifically, two fundamental questions have yet to be fully addressed: (1) what is the trade-off between considering soft penalties versus hard constraints, and (2) what is the statistical gain of incorporating physical knowledge compared to purely data-driven empirical error minimisation. In this paper, we address these questions for PISL in convex classes of functions under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We show that, under suitable assumptions, (1) the error rates of physics-informed estimators are comparable to those of hard constrained empirical error minimisers, differing only by constant terms, and that (2) informed penalization can effectively reduce model complexity, akin to dimensionality reduction, thereby improving learning performance. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in convex classes of functions, contributing to closing the gap between statistical theory and practical PISL, with potential applications to cases not yet explored in the literature.

Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method

TL;DR

This work addresses the statistical properties of physics-informed estimators when the physics is encoded via a linear differential operator. It develops complexity-dependent error bounds using the small-ball method for convex function classes, comparing soft physics-informed penalties to hard constraints and quantifying the statistical gain from incorporating physical knowledge. The key findings show that, under suitable assumptions, the error rates of physics-informed estimators are close to those of hard-constrained empirical minimizers up to constants, and that informed penalization can effectively reduce model complexity by restricting to the kernel of the operator, akin to dimensionality reduction. The results provide a general theoretical framework for evaluating PISL methods beyond specific PDEs, with implications for PINNs and broader physics-informed learning in convex settings.

Abstract

Physics-informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties is still missing. Specifically, two fundamental questions have yet to be fully addressed: (1) what is the trade-off between considering soft penalties versus hard constraints, and (2) what is the statistical gain of incorporating physical knowledge compared to purely data-driven empirical error minimisation. In this paper, we address these questions for PISL in convex classes of functions under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We show that, under suitable assumptions, (1) the error rates of physics-informed estimators are comparable to those of hard constrained empirical error minimisers, differing only by constant terms, and that (2) informed penalization can effectively reduce model complexity, akin to dimensionality reduction, thereby improving learning performance. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in convex classes of functions, contributing to closing the gap between statistical theory and practical PISL, with potential applications to cases not yet explored in the literature.
Paper Structure (23 sections, 7 theorems, 88 equations)

This paper contains 23 sections, 7 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Physics-informed statistical learning
  3. The statistics of physics-informed regularization
  4. Regularization in statistical learning
  5. Small-ball method and complexity dependent error rates
  6. Main contributions
  7. Paper structure
  8. Error Rates for Physics-informed Statistical Learning
  9. Small-ball property of convex-hull
  10. Concentration for target functions
  11. Candidates for $\hat{h}_{n,\Psi}$
  12. Excess loss in the convex hull of ${\mathscr H}$
  13. Local Rademacher complexities
  14. Statement of results
  15. Statistical effect of physics-informed penalization
  16. ...and 8 more sections

Key Result

Theorem 2.4

Let ${\mathscr H}$ be a class of functions such that ${\mathscr F} = \widebar{\text{conv } \mathscr{H}}$ satisfies Assumption small_ball with constants $\kappa$ and $\epsilon$. Fix $\gamma \geq 0$ such that $\mathscr{H}^{\star} = \mathscr{H}^{\star}(\gamma)$ satisfies Assumption concentration. Set and $\lambda > \max\{\lambda_{0}(\delta,\tau),1\}$ with $0 < \tau < \tau_{n}(\rho)$ and $\delta \in

Theorems & Definitions (15)

  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1: Theorem 1.6 of lecue2017regularization
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • proof : Proof of Theorem \ref{['main_theorem']}
  • ...and 5 more