# DPG loss functions for learning parameter-to-solution maps by neural networks

arXiv: https://arxiv.org/abs/2506.18773 · HTML: https://mlsanity.com/paper/2506.18773/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 49 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 15 of 15)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks](https://mlsanity.com/paper/2004.12131/) (2020)
- [Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks](https://mlsanity.com/paper/1904.01514/) (2019)
- [Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2012.08023/) (2020)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [Multilevel CNNs for Parametric PDEs](https://mlsanity.com/paper/2304.00388/) (2023)
- [First Order System Least Squares Neural Networks](https://mlsanity.com/paper/2409.20264/) (2024)

## Built on by (most cited first, 2 of 2)

- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)
- [Global error estimators for parametric monotone nonlinearities and neural approximations](https://mlsanity.com/paper/2610.05767/) (2026)

## Abstract (arXiv)

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 the prediction capability of the 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.