mlsanityData to 7 Oct 2026

Gauss-Newton Natural Gradient Descent for Physics-Informed Computational Fluid Dynamics

arXiv 2402.10680 ↗First posted 16 Feb 2024math.OC physics.comp-phcited by 27

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Abstract (arXiv)

We propose Gauss-Newton's method in function space for the solution of the Navier-Stokes equations in the physics-informed neural network (PINN) framework. Upon discretization, this yields a natural gradient method that provably mimics the function space dynamics. Our computational results demonstrate close to single-precision accuracy measured in relative L2 norm on a number of benchmark problems. To the best of our knowledge, this constitutes the first contribution in the PINN literature that solves the Navier-Stokes equations to this degree of accuracy. Finally, we show that given a suitable integral discretization, the proposed optimization algorithm agrees with Gauss-Newton's method in parameter space. This allows a matrix-free formulation enabling efficient scalability to large network sizes.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Feb 2024.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 11; 2026: 11.

27 papers.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 55
  3. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
  4. 2024Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networkscited by 21
  5. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  6. 2025Improving Energy Natural Gradient Descent through Woodbury, Momentum, and Randomizationcited by 13
  7. 2026Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networkscited by 11
  8. 2024ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learningcited by 7
  9. 2025High precision PINNs in unbounded domains: application to singularity formulation in PDEscited by 6
  10. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  11. 2025Operator Learning at Machine Precisioncited by 6
  12. 2025Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step casecited by 3
15 more
  1. 2025Enforcing hidden physics in physics-informed neural networkscited by 3
  2. 2024A Structure-Guided Gauss-Newton Method for Shallow ReLU Neural Networkcited by 2
  3. 2025Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networkscited by 2
  4. 2026Natural gradient descent with momentumcited by 2
  5. 2025Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebracited by 1
  6. 2026On the Convergence Behavior of Preconditioned Gradient Descent Toward the Rich Learning Regimecited by 1
  7. 2026Error whitening: Why Gauss-Newton outperforms Newtoncited by 1
  8. 2026An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networkscited by 1
  9. 2025Diverse Topology Optimization using Modulated Neural Fieldscited by 0
  10. 2026Gauss-Newton Natural Gradient Descent for Shape Learningcited by 0
  11. 2026Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limitcited by 0
  12. 2026Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solverscited by 0
  13. 2026From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNscited by 0
  14. 2026SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learningcited by 0
  15. 2026Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solverscited by 0