PDGM-PINN: Partial Derivative Guided Multi-Branch Physics-Informed Neural Network

Shangpeng Lei, Chenghan Yang, Roberts Grants, Uldis Grunde, Nadezhda Kunicina

Mathematics, 2026

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DOI 10.3390/math14081349
Data confidence 70 medium
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Abstract

With the development of scientific machine learning (SciML), the proposal of physics-informed neural networks (PINNs) has provided a powerful paradigm for solving partial differential equations (PDEs). While PINNs perform well in solving high-dimensional PDEs, they perform worse than traditional numerical methods for low-dimensional problems. This discrepancy arose from potential convergence conflicts induced by distinct physical magnitude of loss terms. To decouple the convergence conflicts, we propose a partial derivative guided multi-branch physics-informed neural network (PDGM-PINN). Inspired by SciML, we treat both the solution and partial derivatives as dependent variables to be predicted. The partial derivatives are directly predicted by sub-branches, while the main branch approximates the PDE solution, and all branches share error backpropagation information. Furthermore, we redesign the loss function. The loss of the governing equation is computed with the solution and partial derivatives predicted by the main and sub-branches. Schwarz’s theorem and Kullback–Leibler divergence are incorporated into the loss terms as soft constraints of partial derivatives continuity and residual distributions consistency for the governing equations. We conducted comprehensive experimental evaluations on seven PDEs, and ablation experiments, sensitivity analyses, and complexity analyses were carried out to investigate the rationality of PDGM-PINN. The results demonstrate that PDGM-PINN achieves the best performance among PINN variants with the fewest trainable parameters, effectively avoiding architectural redundancy.