mlsanityData to 7 Oct 2026

Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions

arXiv 2407.20741 ↗First posted 30 Jul 2024cs.LG cs.NA math.DScited by 0

Not on the map. Why?

Abstract (arXiv)

"AI for Science" aims to solve fundamental scientific problems using AI techniques. As most physical phenomena can be described as Partial Differential Equations (PDEs) , approximating their solutions using neural networks has evolved as a central component of scientific-ML. Physics-Informed Neural Networks (PINNs) is the general method that has evolved for this task but its training is well-known to be very unstable. In this work we explore the possibility of changing the model being trained from being just a neural network to being a non-linear transformation of it - one that algebraically includes the boundary/initial conditions. This reduces the number of terms in the loss function than the standard PINN losses. We demonstrate that our modification leads to significant performance gains across a range of benchmark tasks, in various dimensions and without having to tweak the training algorithm. Our conclusions are based on conducting hundreds of experiments, in the fully unsupervised setting, over multiple linear and non-linear PDEs set to exactly solvable scenarios, which lends to a concrete measurement of our performance gains in terms of order(s) of magnitude lower fractional errors being achieved, than by standard PINNs. The code accompanying this manuscript is publicly available at, https://github.com/MorganREN/Improving-PINNs-By-Algebraic-Inclusion-of-Boundary-and-Initial-Conditions

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 29 of its 56 references that are in the map. First posted on arXiv 30 Jul 2024.

Builds on

29 of 56 references.

References · 29

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  2. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  5. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  7. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  8. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  9. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
  10. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  11. 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
  12. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
17 more
  1. 2021Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equationscited by 68
  2. 2019Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Predictioncited by 67
  3. 2017Data-driven discovery of Koopman eigenfunctions for controlcited by 64
  4. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  5. 2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 46
  6. 2018Solving the Kolmogorov PDE by means of deep learningcited by 41
  7. 2021Will Artificial Intelligence supersede Earth System and Climate Models?cited by 37
  8. 2020Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networkscited by 27
  9. 2020Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3Dcited by 26
  10. 2020Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equationscited by 24
  11. 2020Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learningcited by 22
  12. 2022Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networkscited by 19
  13. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  14. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  15. 2023Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equationscited by 10
  16. 2023Global Convergence of SGD For Logistic Loss on Two Layer Neural Netscited by 3
  17. 2023Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUpcited by 2

Built on by

Not cited yet.