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Symmetry Constraints Enhance Long-term Stability and Accuracy in Unsupervised Learning of Geophysical Fluid Flows
  • Yunfei Huang,
  • David S. Greenberg
Yunfei Huang
Institute of Coastal Systems, Helmholtz-Zentrum Hereon
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David S. Greenberg
Helmholtz-Zentrum Hereon, Geesthacht, Germany

Corresponding Author:[email protected]

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Fluid dynamical systems are well described by discretized partial differential equations, but computational costs limit accuracy, duration and/or resolution in numerical integrations. Recent studies showed that deep neural networks trained on simulations or PDE-derived losses can improve cost-accuracy tradeoffs, but purely data-centric approaches discard physical and mathematical insights and require computationally costly training data. Here we draw on advances in geometric deep learning to design solver networks that respect PDE symmetries as hard constraints. We construct equivariant convolutional layers for mixed scalar-vector input fields in order to capture the symmetries inherent to specific PDEs. We demonstrate our approach on a challenging 1D semi-implicit shallow water scheme with closed boundaries, applying unsupervised learning with a physics-derived loss function. We report strong improvements in accuracy and stability of equivariant solvers compared to standard convolutional networks with the same architectures and parameter counts. Solver equivariance also improves performance on new initial conditions not encountered during training, and suppresses error accumulation in global momentum and energy. Strikingly, these benefits do not reduce loss values during training, but appear later during ML-assisted rollouts over time steps. Our results suggest that symmetry constraints could improve deep learning performance across a wide range of fluid dynamical tasks, learning algorithms and neural architectures.
05 Feb 2024Submitted to ESS Open Archive
10 Feb 2024Published in ESS Open Archive