Gradient-enhanced PINN with residual unit for studying forward-inverse problems of variable coefficient equations

TL;DR

The paper introduces R-gPINN, a gradient-enhanced PINN with residual units designed to solve forward-inverse problems of variable coefficient PDEs. By incorporating gradient information of the variable coefficient $V(t)$ into the loss and employing residual blocks with pre- or post-activation connections, the method improves both solution accuracy and coefficient discovery relative to PINN and gPINN baselines. Across Burgers, KdV, Sine-Gordon, and KP equations, R-gPINN achieves substantial error reductions in predicted solution $\,\hat{u}$ and learned coefficient $\hat{V}(t)$, with gains varying by problem and residual configuration. The approach demonstrates strong generalization in data-scarce regimes and offers a flexible framework for high-dimensional variable-coefficient PDEs, motivating future work on adaptive sampling and richer physics constraints.

Abstract

Physics-informed neural network (PINN) is a powerful emerging method for studying forward-inverse problems of partial differential equations (PDEs), even from limited sample data. Variable coefficient PDEs, which model real-world phenomena, are of considerable physical significance and research value. This study proposes a gradient-enhanced PINN with residual unit (R-gPINN) method to solve the data-driven solution and function discovery for variable coefficient PDEs. On the one hand, the proposed method incorporates residual units into the neural networks to mitigate gradient vanishing and network degradation, unify linear and nonlinear coefficient problem. We present two types of residual unit structures in this work to offer more flexible solutions in problem-solving. On the other hand, by including gradient terms of variable coefficients, the method penalizes collocation points that fail to satisfy physical properties. This enhancement improves the network's adherence to physical constraints and aligns the prediction function more closely with the objective function. Numerical experiments including solve the forward-inverse problems of variable coefficient Burgers equation, variable coefficient KdV equation, variable coefficient Sine-Gordon equation, and high-dimensional variable coefficient Kadomtsev-Petviashvili equation. The results show that using R-gPINN method can greatly improve the accuracy of predict solution and predict variable coefficient in solving variable coefficient equations.

Figures (18)

• Figure 1: (Color online) (a) Pre-activation residual unit. (b) Post-activation residual unit.
• Figure 2: (Color online) The flowchart of the R-gPINN algorithm for solving forward-inverse problems of variable coefficient PDEs: $\{\partial \mathbf{x}\}$ denotes the gradients with respect to $x$ required during the network training process.
• Figure 3: (Color online) The whole drawing is divided into two parts: the upper part of the figure is the dynamic image to compare the exact solutions and predicted solutions of the variable coefficient Burgers equations, and the errors between them; the lower part displays a time slice at a specific point in time. “ Exact Dynamics" represents the exact values within the solution domain, while “ Learned Dynamics Data" represents the predicted values obtained from R-PINN. Initial and boundary points are indicated by purple “ x" markers.
• Figure 4: (Color online) 3D surface map and contour projections along the axes of the predicted solution for the variable coefficient Burgers equation via R-PINN.
• Figure 5: (Color online) Data-driven discovery of the linear coefficient $v(t) = t$ in the variable coefficient Burgers equation: the upper part of the figure shows the dynamic evolution of the exact solutions, predicted solutions, and their corresponding errors; the lower part displays the time slices at specific points in time.

Theorems & Definitions (1)

• Remark 1