A Feature Fusion Method Based on DeepONet for Dynamic Equations

Dynamic systems governed by differential–algebraic equations (DAEs) play a crucial role in modeling various physical phenomena. However, the computational solution encounters significant challenges due to the inherent rigidity and intricate coupling between the differential and algebraic parts, especially when dealing with high exponents. In this paper, a novel method is presented for effectively solving index-3 form DAEs, employing a combination of multiscale techniques and the physics-informed deep operator neural networks (PiDeepONets). To tackle the computational complexity arising from high indices, the presented approach first transforms the index-3 DAE into index-1 form, reducing stiffness and simplifying the solution process. The integration of the PiDeepONets framework empowers the method with neural network expressiveness while incorporating physics-based constraints, thereby considering both the system's initial conditions and physical laws during the learning process. Furthermore, multi-scale strategy is used to handle scenes with high-frequency or multi-scale features. Experimental validation demonstrates the effectiveness of the proposed method in handling DAE systems.