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Equation identification for fluid flows via physics-informed neural networks

arXiv 2408.17271 ↗First posted 30 Aug 2024physics.flu-dyn cs.LGcited by 0

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Abstract (arXiv)

Scientific machine learning (SciML) methods such as physics-informed neural networks (PINNs) are used to estimate parameters of interest from governing equations and small quantities of data. However, there has been little work in assessing how well PINNs perform for inverse problems across wide ranges of governing equations across the mathematical sciences. We present a new and challenging benchmark problem for inverse PINNs based on a parametric sweep of the 2D Burgers' equation with rotational flow. We show that a novel strategy that alternates between first- and second-order optimization proves superior to typical first-order strategies for estimating parameters. In addition, we propose a novel data-driven method to characterize PINN effectiveness in the inverse setting. PINNs' physics-informed regularization enables them to leverage small quantities of data more efficiently than the data-driven baseline. However, both PINNs and the baseline can fail to recover parameters for highly inviscid flows, motivating the need for further development of PINN methods.

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References · 19

  1. 2015Automatic differentiation in machine learning: a surveycited by 846
  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. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  5. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  6. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  7. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
  8. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  9. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  10. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 86
  11. 2020A fast and accurate physics-informed neural network reduced order model with shallow masked autoencodercited by 80
  12. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
7 more
  1. 2021Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networkscited by 43
  2. 2021Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusioncited by 38
  3. 2022Applications of physics informed neural operatorscited by 33
  4. 2023MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networkscited by 17
  5. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  6. 2023Augmenting a Physics-Informed Neural Network for the 2D Burgers Equation by Addition of Solution Data Pointscited by 5
  7. 2022Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equationscited by 4

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