# PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries

arXiv: https://arxiv.org/abs/2004.06490 · HTML: https://mlsanity.com/paper/2004.06490/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2020.

## Built on by (most cited first, 30 of 38)

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- [BINN: A deep learning approach for computational mechanics problems based on boundary integral equations](https://mlsanity.com/paper/2301.04480/) (2023)
- [CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries](https://mlsanity.com/paper/2110.01359/) (2021)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
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- [Neural PDE Solvers for Irregular Domains](https://mlsanity.com/paper/2211.03241/) (2022)
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- [Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computation](https://mlsanity.com/paper/2405.11208/) (2024)
- [Adaptive deep density approximation for fractional Fokker-Planck equations](https://mlsanity.com/paper/2210.14402/) (2022)
- [The Hard-Constraint PINNs for Interface Optimal Control Problems](https://mlsanity.com/paper/2308.06709/) (2023)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)
- [TSONN: Time-stepping-oriented neural network for solving partial differential equations](https://mlsanity.com/paper/2310.16491/) (2023)
- [An explainable operator approximation framework under the guideline of Green's function](https://mlsanity.com/paper/2412.16644/) (2024)
- [On the Compatibility between Neural Networks and Partial Differential Equations for Physics-informed Learning](https://mlsanity.com/paper/2212.00270/) (2022)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)

## Abstract (arXiv)

We present PFNN, a penalty-free neural network method, to efficiently solve a class of second-order boundary-value problems on complex geometries. To reduce the smoothness requirement, the original problem is reformulated to a weak form so that the evaluations of high-order derivatives are avoided. Two neural networks, rather than just one, are employed to construct the approximate solution, with one network satisfying the essential boundary conditions and the other handling the rest part of the domain. In this way, an unconstrained optimization problem, instead of a constrained one, is solved without adding any penalty terms. The entanglement of the two networks is eliminated with the help of a length factor function that is scale invariant and can adapt with complex geometries. We prove the convergence of the PFNN method and conduct numerical experiments on a series of linear and nonlinear second-order boundary-value problems to demonstrate that PFNN is superior to several existing approaches in terms of accuracy, flexibility and robustness.