🤖 AI Summary
This work addresses the limitations of conventional physics-informed neural networks in solving partial differential equations, which often lack rigorous control over numerical errors and guarantees of robustness and convergence. The authors propose a novel framework based on a discrete variational formulation, wherein function spaces, inner products, and weak forms are defined over discrete point sets. By integrating neural networks with discrete automatic differentiation and employing Kronecker delta test functions, they construct a robust loss functional directly linked to the true approximation error. The method synergistically combines discrete finite-difference derivatives, automatic differentiation, and Adamax optimization to achieve controllable error behavior during training. Numerical experiments demonstrate that the approach exhibits superior convergence, robustness, and effective suppression of numerical errors on benchmark problems, including two-dimensional Stokes and Laplace equations.
📝 Abstract
We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two-dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well-posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.