Physics-informed neural networks with hybrid Kolmogorov-Arnold network and augmented Lagrangian function for solving partial differential equations

Physics-informed neural networks (PINNs) have emerged as a fundamental approach within deep learning for the resolution of partial differential equations (PDEs). Nevertheless, conventional multilayer perceptrons (MLPs) are characterized by a lack of interpretability and encounter the spectral bias problem, which diminishes their accuracy and interpretability when used as an approximation function within the diverse forms of PINNs. Moreover, these methods are susceptible to the over-inflation of penalty factors during optimization, potentially leading to pathological optimization with an imbalance between various constraints. In this study, we are inspired by the Kolmogorov-Arnold network (KAN) to address mathematical physics problems and introduce a hybrid encoder-decoder model to tackle these challenges, termed AL-PKAN. Specifically, the proposed model initially encodes the interdependencies of input sequences into a high-dimensional latent space through the gated recurrent unit (GRU) module. Subsequently, the KAN module is employed to disintegrate the multivariate function within the latent space into a set of trainable univariate activation functions, formulated as linear combinations of B-spline functions for the purpose of spline interpolation of the estimated function. Furthermore, we formulate an augmented Lagrangian function to redefine the loss function of the proposed model, which incorporates initial and boundary conditions into the Lagrangian multiplier terms, rendering the penalty factors and Lagrangian multipliers as learnable parameters that facilitate the dynamic modulation of the balance among various constraint terms. Ultimately, the proposed model exhibits remarkable accuracy and generalizability in a series of benchmark experiments, thereby highlighting the promising capabilities and application horizons of KAN within PINNs.