A MIXED FORMULATION-BASED PHYSICS-INFORMED KOLMOGOROV-ARNOLD NETWORK FOR HETEROGENEOUS SOLID PROBLEMS

Heterogeneous solids exhibit material parameter discontinuities and multiscale characteristics, which pose significant challenges for efficient modeling and accurate simulation. Physics-informed neural networks (PINNs) provide a promising mesh-free framework by embedding physical laws into neural network training. However, classical PINNs may suffer from limited accuracy and efficiency when applied to heterogeneous solid problems. In this work, a mixed formulation-based physics-informed Kolmogorov–Arnold network (MPIKAN) is proposed for solving Poisson and elasticity problems in heterogeneous solids. On the one hand, the proposed method constructs the loss function based on the residuals of the governing equations and boundary conditions in a mixed formulation. By introducing auxiliary variables, the order of derivatives required in the residual terms is reduced, thereby alleviating the computational cost of automatic differentiation, while the continuity constraints across material interfaces are naturally incorporated within the mixed framework. On the other hand, a Kolmogorov–Arnold network (KAN) is employed to approximate both the primary variables and the auxiliary variables. Compared with conventional deep neural networks, KANs have been reported to exhibit reduced spectral bias, which is advantageous for representing localized features and multiscale behaviors commonly observed in heterogeneous solids. The proposed method is applied to a Poisson problem with a circular inclusion and a uniaxial compression problem of a dual-mineral rock. For the circular-inclusion Poisson problem, MPIKAN achieves reliable and accurate solutions, and demonstrates better accuracy and higher parameter efficiency than the mixed formulation-based PINN (MPINN) model. In addition, both MPIKAN and MPINN constructed with mixed formulation-based loss functions show clear advantages over classical strong-form PINNs, for which the interface continuity conditions are not explicitly modeled and the solution accuracy may deteriorate significantly in the presence of material discontinuities. The proposed MPIKAN model is further applied to the uniaxial compression problem of a dual-mineral rock characterized by more complex heterogeneous microstructures, and delivers accurate predictions. Overall, the above results demonstrate the effectiveness of the proposed MPIKAN model, highlighting its potential for efficient and robust simulation of heterogeneous solid problems.