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Convergence Properties of PINNs for the Navier-Stokes-Cahn-Hilliard System

Kevin Buck, Roger Temam

TL;DR

The paper introduces a simplified a priori framework to analyze the convergence properties of Physics-Informed Neural Networks (PINNs) for PDEs, focusing on the Cahn–Hilliard and Navier–Stokes–Cahn–Hilliard (NSCH) systems. It formalizes consistency and convergence notions in an operator setting, deriving explicit error bounds that relate the residual loss to the approximation error, including a problem-dependent convergence order and a condition number. For Cahn–Hilliard, a rigorous bound is established showing at least first-order convergence with an exponential-in-time factor and constants tied to the framework's regularization and loss weights. For the NSCH system, a consistency result is proved, complemented by numerical experiments on a toy problem that estimate a convergence rate around 1.15 and a modest condition number, providing practical insight into loss-driven accuracy. The work offers a predictive lens for PINN performance on phase-field and two-phase flow problems and motivates further refinement through broader benchmarks and parameter-dependent analyses.

Abstract

Approximating solutions to differential equations using neural networks has become increasingly popular and shows significant promise. In this paper, we propose a simplified framework for analyzing the potential of neural networks to simulate differential equations based on the properties of the equations themselves. We apply this framework to the Cahn-Hilliard and Navier-Stokes-Cahn-Hilliard systems, presenting both theoretical analysis and practical implementations. We then conduct numerical experiments on toy problems to validate the framework's efficacy in accurately capturing the desired properties of these systems and numerically estimate relevant convergence properties.

Convergence Properties of PINNs for the Navier-Stokes-Cahn-Hilliard System

TL;DR

The paper introduces a simplified a priori framework to analyze the convergence properties of Physics-Informed Neural Networks (PINNs) for PDEs, focusing on the Cahn–Hilliard and Navier–Stokes–Cahn–Hilliard (NSCH) systems. It formalizes consistency and convergence notions in an operator setting, deriving explicit error bounds that relate the residual loss to the approximation error, including a problem-dependent convergence order and a condition number. For Cahn–Hilliard, a rigorous bound is established showing at least first-order convergence with an exponential-in-time factor and constants tied to the framework's regularization and loss weights. For the NSCH system, a consistency result is proved, complemented by numerical experiments on a toy problem that estimate a convergence rate around 1.15 and a modest condition number, providing practical insight into loss-driven accuracy. The work offers a predictive lens for PINN performance on phase-field and two-phase flow problems and motivates further refinement through broader benchmarks and parameter-dependent analyses.

Abstract

Approximating solutions to differential equations using neural networks has become increasingly popular and shows significant promise. In this paper, we propose a simplified framework for analyzing the potential of neural networks to simulate differential equations based on the properties of the equations themselves. We apply this framework to the Cahn-Hilliard and Navier-Stokes-Cahn-Hilliard systems, presenting both theoretical analysis and practical implementations. We then conduct numerical experiments on toy problems to validate the framework's efficacy in accurately capturing the desired properties of these systems and numerically estimate relevant convergence properties.
Paper Structure (14 sections, 3 theorems, 125 equations, 3 figures)

This paper contains 14 sections, 3 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Description of the Neural Network
  3. Physics-Informed Neural Networks
  4. Analytical Concerns
  5. Analytical Framework
  6. Language of Operators
  7. Cahn-Hilliard
  8. The Cahn-Hilliard Physics-Informed Neural Network
  9. Results for Cahn-Hilliard
  10. Navier-Stokes-Cahn-Hilliard
  11. Navier-Stokes Cahn-Hillard PINN
  12. Results for Navier-Stokes-Cahn Hilliard
  13. Numerical Results
  14. Conclusion and Further Thoughts

Key Result

Theorem 2.1

Let $\phi_\theta(x,t)$ be the neural network with outputs in [-1, 1] and $\phi(x,t)$ a weak, physical solution to the Cahn-Hilliard equations (2). If the Loss associated with the network, defined by (Loss1), satisfies then Where $\lambda > 0$ can be chosen arbitrarily. $C_\lambda$ depends only on $\lambda$ and $\Omega$. Additionally $C_\lambda\searrow C$ as $\lambda \nearrow\infty$ and $C_\lambd

Figures (3)

  • Figure 1: We see the loss decreasing on an arbitrarily chosen problem different from our other experiments. For this problem, the Loss (without the 1000 weight on the initial condition) is on the order of $3*10^{-3}$
  • Figure 2: The $L^2$ error at the final timestep plotted against the Emperical Loss Value for the described PINN modeling the NSCH system given above, before any processing of data. We can see that the actual $L^2$ error of the simulation is quite small for small Loss values, implying the desired convergence. In some cases the error is as small as about $10^{-4}$
  • Figure 3: The $L^2$ error vs the Emperical Loss Value for the described PINN modeling the NSCH system, after the processing of data to a base 10 logscale. We can see that the data descends roughly in parallel with the derived line of best fit with condition number $.09$ and rate of convergence $1.153$

Theorems & Definitions (9)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Definition 5.1
  • Theorem 5.2
  • proof