A quantum-inspired deep neural network framework for physically constrained PDEs

We propose a novel quantum-inspired deep neural network framework (QIDNNF) for solving partial differential equations (PDEs), specifically Schrödinger equation and those derived through Schrödingerization. QIDNNF integrates fundamental quantum mechanics principles, including global phase invariance and normalization, to ensure unitary quantum dynamics and the preservation of conservation laws. Through numerical experiments, QIDNNF exhibits superior stability over finite difference schemes for large time steps, improved long-term accuracy over neural ordinary differential equations (Neural ODEs) and physics-informed neural networks (PINNs), and predictive precision unaffected by variations in initial phase angles. Furthermore, QIDNNF effectively models real-world physical systems, including 1D nonlinear wave propagation and 2D and 3D flow evolution, demonstrating their accuracy and consistency in simulating complex physical phenomena.