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An operator preconditioning perspective on training in physics-informed machine learning

Tim De Ryck, Florent Bonnet, Siddhartha Mishra, Emmanuel de Bézenac

TL;DR

This work analyzes why physics-informed machine learning models, such as PINNs, can be slow to train and shows that training speed is governed by the conditioning of the Hermitian square of the underlying differential operator combined with a kernel integral operator. The authors derive a linearized gradient-descent dynamics and prove convergence rates in terms of the condition number of an operator $\mathcal{A} = \mathcal{D}^{*}\mathcal{D}$ composed with the kernel operator $TT^{*}$, linking optimization performance to spectral properties. They propose explicit operator-preconditioning strategies via linear transforms of parameters or gradients to approximate $\mathcal{A}^{-1}$ and demonstrate their effectiveness on Poisson, Helmholtz, and linear advection problems, including Fourier-feature models. The results provide a principled framework for understanding and improving PINN training, connecting existing heuristics such as boundary-condition handling and domain decomposition to operator conditioning and opening avenues for nonlinear preconditioning in future work.

Abstract

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

An operator preconditioning perspective on training in physics-informed machine learning

TL;DR

This work analyzes why physics-informed machine learning models, such as PINNs, can be slow to train and shows that training speed is governed by the conditioning of the Hermitian square of the underlying differential operator combined with a kernel integral operator. The authors derive a linearized gradient-descent dynamics and prove convergence rates in terms of the condition number of an operator composed with the kernel operator , linking optimization performance to spectral properties. They propose explicit operator-preconditioning strategies via linear transforms of parameters or gradients to approximate and demonstrate their effectiveness on Poisson, Helmholtz, and linear advection problems, including Fourier-feature models. The results provide a principled framework for understanding and improving PINN training, connecting existing heuristics such as boundary-condition handling and domain decomposition to operator conditioning and opening avenues for nonlinear preconditioning in future work.

Abstract

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.
Paper Structure (42 sections, 7 theorems, 60 equations, 20 figures, 2 tables)

This paper contains 42 sections, 7 theorems, 60 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Analyzing training for physics-informed machine learning in terms of operator conditioning.
  3. Setting.
  4. Characterization of Gradient Descent for Physics-informed Machine Learning.
  5. Convergence of Simplified Gradient Descent Iterations (\ref{['eq:SGD']}).
  6. Operator Conditioning.
  7. Preconditioning and improving training in Physics-informed machine learning.
  8. Explicit preconditioning by linearly transforming parameters.
  9. Explicit preconditioning by linearly transforming the gradients.
  10. Analysis of the impact of preconditioning for the Poisson equation.
  11. Viewing available strategies for improving training in physics-informed machine learning models through the lens of operator (pre-)conditioning.
  12. Discussion.
  13. Summary.
  14. Related Work.
  15. Limitations and Future Work.
  16. ...and 27 more sections

Key Result

Lemma 2.1

Let $\delta>0$ be such that $\max_k \|\varepsilon_k\|_2 \leq \delta$. If ${\mathbb{A}}$ is invertible and $\eta=c/\max_j \abs{\lambda_j({\mathbb{A}})}$ for some $0<c<1$ then it holds for any $k\in\mathbb{N}$ that,

Figures (20)

  • Figure 1: Poisson equation with Fourier features. Left: Optimal condition number vs. Number of Fourier features. Right: Training for the unpreconditioned and preconditioned Fourier features.
  • Figure 2: Linear advection equation with Fourier features. Left: Optimal condition number vs. $\beta$. Right: Training for the unpreconditioned and preconditioned Fourier features.
  • Figure 3: Evolution of condition number of preconditioned matrix for different $\gamma$ and $\lambda$ (left) and for different $\gamma$ for the optimal $\lambda$ (right).
  • Figure 4: Left: optimal $\lambda$ in terms of $K$. Right: Evolution of condition number of ${\mathbb{A}}(\lambda)$ in terms of model size $K$ for multiple choices of $\lambda$.
  • Figure 5: Evolution of condition number of ${\mathbb{A}}'$ for soft boundary conditions (full lines) and hard boundary conditions (dotted lines) in terms of $\beta$ and $\lambda$ for $K=3$
  • ...and 15 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Lemma A.1
  • proof
  • ...and 5 more