A physics-informed hybrid architecture of alternating KAN and MLP for ill-posed nonlinear wave equations with unstable solutions

In this paper, we introduce a hybrid neural network framework, alternating Kolmogorov-Arnold Network and Multi-Layer Perceptron (A-KMNet), which synergistically combines the advantages of Multi-Layer Perceptrons (MLPs) and Kolmogorov-Arnold Networks (KANs) for solving nonlinear wave equations. Based on the initial layer in the alternating stack, the A-KMNet framework is categorized into two architectures: a KAN-based variant, denoted as A-KMNetKAN, and an MLP-based variant, A-KMNetMLP. We develop their corresponding physics-informed variants, Physics-Informed A-KMNetMLP (PI-A-KMNetMLP) and Physics-Informed A-KMNetKAN (PI-A-KMNetKAN), and apply them to solve the ill-posed 'bad' Boussinesq and Jaulent-Miodek (JM) equations. These two 'bad' equations pose significant challenges for traditional grid-based numerical methods, particularly for linearly unstable small-amplitude waves. Our comparative analysis against Physics-Informed Neural Networks (PINNs) and Physics-Informed KANs (PI-KANs) reveals that while all methods achieve high precision for simple solutions, PI-A-KMNetKAN offers a distinct accuracy advantage. However, for complex dynamics with a large number of trainable parameters, MLP-based architectures significantly outperform their KAN-based counterparts by one to two orders of magnitude, highlighting their superior scalability. The A-KMNet framework successfully captures wave evolution in regimes previously considered intractable. Furthermore, for inverse problems, both A-KMNet variants demonstrate robust parameter inference capabilities. Given that different architectures exhibit complementary strengths in identifying specific parameters, we recommend a multi-method ensemble approach for reliable cross-validation in multi-parameter inversion tasks.