DPG loss functions for learning parameter-to-solution maps by neural networks
This work addresses the challenge of improving prediction capability and robustness in deep neural network reduced models for PDEs, particularly in high-contrast scenarios, but it is incremental as it builds on existing DPG formulations.
The authors tackled the problem of learning parameter-to-solution maps for parameter-dependent PDEs by developing residual-based loss functions, specifically using ultraweak Discontinuous Petrov Galerkin (DPG) discretizations, to enhance accuracy certification and robustness. They demonstrated that for high-contrast diffusion parameters, the DPG loss functions provide much more robust performance than simpler least-squares losses, as shown through numerical results and theoretical arguments.
We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance prediction capability of resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of {high-contrast} diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.