Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equations

TL;DR

The paper tackles the instability in PINNs when solving inverse PDE problems, where data losses remain high while PDE residuals wash out early in training. It introduces Data-guided PINNs (DG-PINNs), a two-phase approach with a data-only pre-training phase followed by a physics-informed fine-tuning phase, aided by an adaptive loss weighting strategy. Across the heat, wave, Euler–Bernoulli beam, and Navier–Stokes equations, DG-PINNs achieve comparable or superior accuracy while significantly reducing training time (approximately 6–9× faster than PINNs) and showing robustness to noisy data. This approach provides a practical, scalable path for efficient inverse PDE solving and can be extended to physics-informed neural operators and broader physical phenomena.

Abstract

Physics-informed neural networks (PINNs) represent a significant advancement in scientific machine learning by integrating fundamental physical laws into their architecture through loss functions. PINNs have been successfully applied to solve various forward and inverse problems in partial differential equations (PDEs). However, a notable challenge can emerge during the early training stages when solving inverse problems. Specifically, data losses remain high while PDE residual losses are minimized rapidly, thereby exacerbating the imbalance between loss terms and impeding the overall efficiency of PINNs. To address this challenge, this study proposes a novel framework termed data-guided physics-informed neural networks (DG-PINNs). The DG-PINNs framework is structured into two distinct phases: a pre-training phase and a fine-tuning phase. In the pre-training phase, a loss function with only the data loss is minimized in a neural network. In the fine-tuning phase, a composite loss function, which consists of the data loss, PDE residual loss, and, if available, initial and boundary condition losses, is minimized in the same neural network. Notably, the pre-training phase ensures that the data loss is already at a low value before the fine-tuning phase commences. This approach enables the fine-tuning phase to converge to a minimal composite loss function with fewer iterations compared to existing PINNs. To validate the effectiveness, noise-robustness, and efficiency of DG-PINNs, extensive numerical investigations are conducted on inverse problems related to several classical PDEs, including the heat equation, wave equation, Euler--Bernoulli beam equation, and Navier--Stokes equation. The numerical results demonstrate that DG-PINNs can accurately solve these inverse problems and exhibit robustness against noise in training data.

Figures (10)

• Figure 1: Schematic of PINNs for solving inverse problems in PDEs.
• Figure 2: Schematic of DG-PINNs for solving inverse problems in PDEs with two phases: the pre-training and fine-tuning phases.
• Figure 3: Sensitivity analysis on $M_{1}$ for the heat equation: (a) the relative $L^{2}$ errors between the testing data and the corresponding prediction from DG-PINN models, and (b) the absolute percentage error (APE) between $\tilde{\beta}^{2}$ and $\beta^{2}$.
• Figure 4: Sensitivity analysis on $N_{d}$ for the heat equation: (a) the relative $L^{2}$ errors between the testing data and the corresponding prediction from DG-PINN models, and (b) the absolute percentage error (APE) between $\tilde{\beta}^{2}$ and $\beta^{2}$.
• Figure 5: Sensitivity analysis on $M_{1}$ for the wave equation: (a) the relative $L^{2}$ errors between the testing data and the corresponding predictions from the DG-PINN models, and (b) the absolute percentage error (APE) between $\tilde{c}^{2}$ and $c^{2}$.