HyResPINNs: Hybrid Residual Networks for Adaptive Neural and RBF Integration in Solving PDEs

TL;DR

HyResPINNs introduce adaptive hybrid residual blocks that combine radial basis function (RBF) networks with standard neural networks (NNs) inside a PINN framework to better capture both smooth and non-smooth PDE solutions. Each residual block dynamically balances RBF and NN contributions via a learnable gating mechanism and uses a regularized, convex-combination formulation to improve training stability. The approach is validated across Allen-Cahn, Darcy flow (smooth and rough coefficients), and Kuramoto–Sivashinsky benchmarks, showing improved accuracy and robustness over standard PINNs and some state-of-the-art baselines, though performance can vary in highly chaotic regimes. By integrating local adaptivity and global smoothness, HyResPINNs offer a principled path toward PDE solvers that leverage the strengths of classical numerical methods and modern deep learning, with potential for better handling multi-scale and discontinuous phenomena in engineering and physics contexts.

Abstract

Physics-informed neural networks (PINNs) have emerged as a powerful approach for solving partial differential equations (PDEs) by training neural networks with loss functions that incorporate physical constraints. In this work, we introduce HyResPINNs, a novel class of PINNs featuring adaptive hybrid residual blocks that integrate standard neural networks and radial basis function (RBF) networks. A distinguishing characteristic of HyResPINNs is the use of adaptive combination parameters within each residual block, enabling dynamic weighting of the neural and RBF network contributions. Our empirical evaluation of a diverse set of challenging PDE problems demonstrates that HyResPINNs consistently achieve superior accuracy to baseline methods. These results highlight the potential of HyResPINNs to bridge the gap between classical numerical methods and modern machine learning-based solvers, paving the way for more robust and adaptive approaches to physics-informed modeling.

Figures (10)

• Figure 1: Illustration of the hybrid residual block with trainable strength connections between the RBF and NN outputs.
• Figure 2: Illustration of the HyResPINN architecture using three hybrid residual blocks. The trainable skip connections $\phi(\beta^{(l)})$ modulate the information flow between blocks.
• Figure 3: 2D Allen-Cahn equation: Comparison of two random RBF kernels from a standard RBF-Net (top row) and the first block of a HyResPINNs (bottom row). Each subplot shows the (final) learned RBF scale parameter ($\tau$) for the selected center points (marked by red crosses).
• Figure 4: Allen-Cahn equation: Comparison of the predicted solutions for the Allen-Cahn equation using HyResPINN, standard PINN, and RBF network models. The top row shows the predicted solutions for HyResPINN (left), standard PINN (center), and RBF PINN (right). The second row shows the absolute error between the predicted and true solutions. The bottom row shows the predicted solutions for time steps ($t=0.25, 0.5, 0.99$) compared to the true solution.
• Figure 5: Allen-Cahn equation: Comparison of the mean relative $L^2$ error using various methods as a function of the number of hidden layers.