🤖 AI Summary
This work addresses the learning of solution operators for nonlinear partial differential equations (PDEs). Methodologically, it introduces a multilayer perceptron (MLP)-based parametric operator learning framework driven by physical inputs—including boundary conditions, coefficients, and source terms—where a local energy functional, constructed via finite element discretization, is directly adopted as the training loss—a novel formulation. To enhance scalability, the method incorporates element-wise parallelization and randomized sparse grid sampling, significantly improving training efficiency for large-scale problems. Experiments on multiple nonlinear PDE benchmarks demonstrate high-accuracy solution prediction and strong generalization across unseen parameter configurations. Compared to conventional numerical solvers, the approach drastically reduces computational overhead associated with repeated parameter sweeps and enables real-time parametric response prediction.
📝 Abstract
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations. The approach is based on a finite element discretization and aims at representing the solution operator by an MLP that takes latent variables as input. The latent variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The loss function is most often an energy functional and we formulate efficient parallelizable training algorithms based on assembling the energy locally on each element. For large problems, the learning process can be made more efficient by using only a small fraction of randomly chosen elements in the mesh in each iteration. The approach is evaluated on several relevant test cases, where learning the solution operator turns out to be beneficial compared to classical numerical methods.