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Hard constraint learning approaches with trainable influence functions for evolutionary equations

Yushi Zhang, Shuai Su, Yong Wang, Yanzhong Yao

TL;DR

The paper addresses the difficulty of training PINNs over long time horizons for evolutionary PDEs by introducing Trainable Hard Constraint PINNs (THC-PINNs) that integrate sequential time-domain learning with influence-function–based hard constraints. By incorporating trainable influence and adjoint functions, along with an adaptive temporal partitioning algorithm, the approach enforces continuity and causality across interval boundaries while adapting to different equation types. The authors demonstrate improved accuracy and stability across convection, Allen–Cahn, KdV, and a 3D heat problem, and show that trainable parameters in the influence functions can significantly enhance performance, particularly for hyperbolic-like dynamics. The method provides a scalable, generalizable framework for long-time evolution of PDEs, with practical benefits for high-dimensional and multi-scale problems, supported by code and data availability.

Abstract

This paper develops a novel deep learning approach for solving evolutionary equations, which integrates sequential learning strategies with an enhanced hard constraint strategy featuring trainable parameters, addressing the low computational accuracy of standard Physics-Informed Neural Networks (PINNs) in large temporal domains.Sequential learning strategies divide a large temporal domain into multiple subintervals and solve them one by one in a chronological order, which naturally respects the principle of causality and improves the stability of the PINN solution. The improved hard constraint strategy strictly ensures the continuity and smoothness of the PINN solution at time interval nodes, and at the same time passes the information from the previous interval to the next interval, which avoids the incorrect/trivial solution at the position far from the initial time. Furthermore, by investigating the requirements of different types of equations on hard constraints, we design a novel influence function with trainable parameters for hard constraints, which provides theoretical and technical support for the effective implementations of hard constraint strategies, and significantly improves the universality and computational accuracy of our method. In addition, an adaptive time-domain partitioning algorithm is proposed, which plays an important role in the application of the proposed method as well as in the improvement of computational efficiency and accuracy. Numerical experiments verify the performance of the method. The data and code accompanying this paper are available at https://github.com/zhizhi4452/HCS.

Hard constraint learning approaches with trainable influence functions for evolutionary equations

TL;DR

The paper addresses the difficulty of training PINNs over long time horizons for evolutionary PDEs by introducing Trainable Hard Constraint PINNs (THC-PINNs) that integrate sequential time-domain learning with influence-function–based hard constraints. By incorporating trainable influence and adjoint functions, along with an adaptive temporal partitioning algorithm, the approach enforces continuity and causality across interval boundaries while adapting to different equation types. The authors demonstrate improved accuracy and stability across convection, Allen–Cahn, KdV, and a 3D heat problem, and show that trainable parameters in the influence functions can significantly enhance performance, particularly for hyperbolic-like dynamics. The method provides a scalable, generalizable framework for long-time evolution of PDEs, with practical benefits for high-dimensional and multi-scale problems, supported by code and data availability.

Abstract

This paper develops a novel deep learning approach for solving evolutionary equations, which integrates sequential learning strategies with an enhanced hard constraint strategy featuring trainable parameters, addressing the low computational accuracy of standard Physics-Informed Neural Networks (PINNs) in large temporal domains.Sequential learning strategies divide a large temporal domain into multiple subintervals and solve them one by one in a chronological order, which naturally respects the principle of causality and improves the stability of the PINN solution. The improved hard constraint strategy strictly ensures the continuity and smoothness of the PINN solution at time interval nodes, and at the same time passes the information from the previous interval to the next interval, which avoids the incorrect/trivial solution at the position far from the initial time. Furthermore, by investigating the requirements of different types of equations on hard constraints, we design a novel influence function with trainable parameters for hard constraints, which provides theoretical and technical support for the effective implementations of hard constraint strategies, and significantly improves the universality and computational accuracy of our method. In addition, an adaptive time-domain partitioning algorithm is proposed, which plays an important role in the application of the proposed method as well as in the improvement of computational efficiency and accuracy. Numerical experiments verify the performance of the method. The data and code accompanying this paper are available at https://github.com/zhizhi4452/HCS.

Paper Structure

This paper contains 17 sections, 34 equations, 15 figures, 12 tables.

Table of Contents

  1. Introduction
  2. A novel hard constraint strategy-based PINN method for solving evolutionary equations
  3. Preliminaries
  4. Standard PINN method
  5. A hard-constrained PINN method that strictly satisfies the initial conditions
  6. Hard constraint formulation for initial conditions with influence functions
  7. Temporal domain segmentation learning strategy based on novel hard constraint
  8. Hard constraint learning strategy for partitioning the temporal domain into two intervals
  9. Hard constraint learning strategy for partitioning the temporal domain into multiple intervals
  10. Adjustable influence and adjoint functions
  11. Adaptive temporal domain partitioning algorithm
  12. Numerical experiments
  13. Convection equation
  14. Allen-Cahn equation
  15. Korteweg-de Vries equation
  16. ...and 2 more sections

Figures (15)

  • Figure 2.1: Influence function and adjoint function using different types of function.
  • Figure 2.2: Influence functions and adjoint functions with different $p_i$.
  • Figure 4.1: The $L_2$ relative errors with different $p_i$ for the convection equation.
  • Figure 4.2: Comparison of the PINN solutions obtained with different $p_i$ and the exact solution $(T=2)$.
  • Figure 4.3: The influence function $\lambda(t,p_i)|_{p_i=1.185}$.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2