mlsanityData to 14 Oct 2026

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

arXiv 2104.08426 ↗First posted 17 Apr 2021math.NA cs.NA cs.NEcited by 137

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Abstract (arXiv)

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artifical neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct φ, an approximate distance function to the boundary of a domain. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as φ multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. We present numerical solutions for linear and nonlinear boundary-value problems over domains with affine and curved boundaries. Benchmark problems in 1D for linear elasticity, advection-diffusion, and beam bending; and in 2D for the Poisson equation, biharmonic equation, and the nonlinear Eikonal equation are considered. The approach extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the 4D hypercube. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

Key facts. Cited by 137 papers in the map (data through 14 Oct 2026). First posted on arXiv 17 Apr 2021.

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Citing papers per year: 2021: 6; 2022: 14; 2023: 20; 2024: 34; 2025: 32; 2026: 31.

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  1. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  2. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  3. 2024The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learningcited by 145
  4. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 126
  5. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  6. 2023Residual-based attention in physics-informed neural networkscited by 72
  7. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  8. 2021Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domainscited by 37
  9. 2022Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)cited by 35
  10. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  11. 2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
  12. 2024Scaling physics-informed hard constraints with mixture-of-expertscited by 24
38 more
  1. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
  2. 2022Improved Training of Physics-Informed Neural Networks with Model Ensemblescited by 19
  3. 2023Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networkscited by 18
  4. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  5. 2024Diffeomorphism Neural Operator for various domains and parameters of partial differential equationscited by 15
  6. 2024Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networkscited by 15
  7. 2022Physics-Informed Neural Networks for Shell Structurescited by 13
  8. 2024Learning in PINNs: Phase transition, diffusion equilibrium, and generalizationcited by 13
  9. 2022Calibrating constitutive models with full-field data via physics informed neural networkscited by 12
  10. 2024Deep Learning Alternatives of the Kolmogorov Superposition Theoremcited by 12
  11. 2023BINN: A deep learning approach for computational mechanics problems based on boundary integral equationscited by 11
  12. 2021CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometriescited by 10
  13. 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
  14. 2021Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learningscited by 9
  15. 2023Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating methodcited by 9
  16. 2024Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networkscited by 9
  17. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  18. 2024BiLO: Bilevel Local Operator Learning for PDE Inverse Problemscited by 8
  19. 2025Solving engineering eigenvalue problems with neural networks using the Rayleigh quotientcited by 8
  20. 2023Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equationscited by 7
  21. 2024Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Basescited by 7
  22. 2022Neural PDE Solvers for Irregular Domainscited by 6
  23. 2023Variational Barycentric Coordinatescited by 6
  24. 2023PINN surrogate of Li-ion battery models for parameter inference. Part I: Implementation and multi-fidelity hierarchies for the single-particle modelcited by 6
  25. 2024Physics-informed Discretization-independent Deep Compositional Operator Networkcited by 6
  26. 2024Robust Biharmonic Skinning Using Geometric Fieldscited by 6
  27. 2025Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddingscited by 6
  28. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  29. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  30. 2023DCEM: A deep complementary energy method for solid mechanicscited by 5
  31. 2023Physics-informed neural wavefields with Gabor basis functionscited by 5
  32. 2024Learning Domain-Independent Green's Function For Elliptic Partial Differential Equationscited by 5
  33. 2024Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applicationscited by 5
  34. 2022Adaptive deep density approximation for fractional Fokker-Planck equationscited by 4
  35. 2023A physics-informed neural network framework for modeling obstacle-related equationscited by 4
  36. 2023The Hard-Constraint PINNs for Interface Optimal Control Problemscited by 4
  37. 2024Energy-based physics-informed neural network for frictionless contact problems under large deformationcited by 4
  38. 2025AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decompositioncited by 4

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