Variational Physics-Informed Neural Network with Functional Constraints Based on Operator Self-Adjointness

Abstract

Traditional Physics-Informed Neural Networks (PINNs) incorporate partial differential equations (PDEs), boundary conditions, and initial conditions into model training. However, their application to multidimensional problems may involve high computational costs associated with high-order automatic differentiation and increasingly demanding sampling requirements as dimensionality grows, potentially limiting the accuracy of solution approximation. To address these issues, we propose a Functional Constraint-based Variational Physics-Informed Neural Network (FC-VPINN) for solving PDEs admitting self-adjoint or weighted self-adjoint representations. It reformulates strong-form PDE constraints as corresponding variational constraints, improving mathematical interpretability while reducing reliance on high-order automatic differentiation. The stationarity condition of the constructed quadratic variational functional is mathematically equivalent to the governing equation under the prescribed conditions. For the second-order PDEs considered in this work, the resulting variational objective involves at most first-order derivatives of the network output, thereby reducing the derivative order required during training. Boundary and initial conditions are imposed through corresponding loss terms, without requiring a predefined set of test functions. Experiments on two-dimensional advection-diffusion, three-dimensional diffusion, and three-dimensional Poisson equations show that FC-VPINN achieves approximately one-order-of-magnitude lower prediction errors than the traditional PINN and reduces memory usage to 40% of that required by the baseline, demonstrating improved accuracy and computational efficiency in multidimensional problems.