mlsanityData to 7 Oct 2026

About optimal loss function for training physics-informed neural networks under respecting causality

arXiv 2304.02282 ↗First posted 5 Apr 2023math.NA cs.AI cs.NAcited by 6

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Abstract (arXiv)

A method is presented that allows to reduce a problem described by differential equations with initial and boundary conditions to the problem described only by differential equations. The advantage of using the modified problem for physics-informed neural networks (PINNs) methodology is that it becomes possible to represent the loss function in the form of a single term associated with differential equations, thus eliminating the need to tune the scaling coefficients for the terms related to boundary and initial conditions. The weighted loss functions respecting causality were modified and new weighted loss functions based on generalized functions are derived. Numerical experiments have been carried out for a number of problems, demonstrating the accuracy of the proposed methods.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 29 of its 66 references that are in the map. First posted on arXiv 5 Apr 2023.

Builds on

29 of 66 references.

References · 29

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  3. 2022Training language models to follow instructions with human feedbackcited by 15,712
  4. 2021Zero-Shot Text-to-Image Generationcited by 4,276
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  6. 2019Mastering Atari, Go, Chess and Shogi by Planning with a Learned Modelcited by 1,327
  7. 2021Transformers in Vision: A Surveycited by 1,009
  8. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  9. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  10. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  11. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  12. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
17 more
  1. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  2. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  3. 2019Learning Deep Transformer Models for Machine Translationcited by 309
  4. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  5. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  6. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  7. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  8. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 122
  9. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 86
  10. 2018Tensor2Tensor for Neural Machine Translationcited by 85
  11. 2019Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transportcited by 60
  12. 2021Adversarial Text-to-Image Synthesis: A Reviewcited by 52
  13. 2020nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applicationscited by 36
  14. 2023ViTO: Vision Transformer-Operatorcited by 29
  15. 2022Meta-PDE: Learning to Solve PDEs Quickly Without a Meshcited by 18
  16. 2022Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)cited by 9
  17. 2023Temporal Consistency Loss for Physics-Informed Neural Networkscited by 3

Built on by

Cited by year

Citing papers per year: 2023: 1; 2024: 4; 2025: 1.

6 papers.

  1. 2023Residual-based attention in physics-informed neural networkscited by 72
  2. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  3. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 55
  4. 2024Separable Physics-Informed Neural Networks for the solution of elasticity problemscited by 5
  5. 2024Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?cited by 3
  6. 2024About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networkscited by 0