# VarNet: Variational Neural Networks for the Solution of Partial Differential Equations

arXiv: https://arxiv.org/abs/1912.07443 · HTML: https://mlsanity.com/paper/1912.07443/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2019.

## Built on by (most cited first, 30 of 35)

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- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries](https://mlsanity.com/paper/2404.12063/) (2024)
- [Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computation](https://mlsanity.com/paper/2405.11208/) (2024)
- [ParticleWNN: a Novel Neural Networks Framework for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.12433/) (2023)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)
- [Machine-learning custom-made basis functions for partial differential equations](https://mlsanity.com/paper/2111.05307/) (2021)
- [A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks](https://mlsanity.com/paper/2211.03627/) (2022)
- [DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods](https://mlsanity.com/paper/2503.10021/) (2025)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.09930/) (2021)
- [Improving hp-Variational Physics-Informed Neural Networks for Steady-State Convection-Dominated Problems](https://mlsanity.com/paper/2411.09329/) (2024)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [Towards Robust Surrogate Models: Benchmarking Machine Learning Approaches to Expediting Phase Field Simulations of Brittle Fracture](https://mlsanity.com/paper/2507.07237/) (2025)
- [Nonlinear discretizations and Newton's method: characterizing stationary points of regression objectives](https://mlsanity.com/paper/2510.11987/) (2025)
- [Visualizing the loss landscapes of physics-informed neural networks](https://mlsanity.com/paper/2602.05849/) (2026)
- [PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs](https://mlsanity.com/paper/2607.20378/) (2026)
- [Path Signatures and Graph Neural Networks for Slow Earthquake Analysis: Better Together?](https://mlsanity.com/paper/2402.03558/) (2024)
- [An efficient hp-Variational PINNs framework for incompressible Navier-Stokes equations](https://mlsanity.com/paper/2409.04143/) (2024)
- [Convex Physics Informed Neural Networks for the Monge-Amp`ere Optimal Transport Problem](https://mlsanity.com/paper/2501.10162/) (2025)

## Abstract (arXiv)

In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. Given this loss function, we also propose an approach to optimally select the space-time samples, used to train the NN, that is based on the feedback provided from the PDE residual. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction (MOR) of PDEs. We demonstrate the performance of our method through extensive numerical experiments for the advection-diffusion PDE as an important case-study.