Local sparse methods improve fluid flow predictions using kernel learning
Local and Sparse Kernel Operator Learning for Incompressible Flows: An Operator Lagrange Formulation
Machine Learning
Summary
Simulating how fluids move and flow is important but hard because it needs to follow special rules like incompressibility. The authors developed a new way to learn these fluid behaviors using local and sparse mathematical tools that keep these rules intact. Their method uses small, manageable pieces that work together, and can run fast on GPUs. This approach reduces errors and storage compared to earlier methods and works well on many fluid flow examples.
What this means in practice
- •For computational fluid dynamics engineers: Generate accurate, scalable models for simulating incompressible fluid flows efficiently using sparse operators on GPUs.
- •For weather prediction teams: Improve fluid flow components in forecasting models by integrating locally learned divergence-free operators for better physical consistency.
Authors
Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar
Abstract
We present a local and sparse kernel method for learning the solution operators of incompressible flows. It builds on our earlier property-preserving kernel method, whose global output reconstruction and dense operator regression scale poorly. Our method replaces the global output reconstruction by local divergence-free interpolants with divergence-free polynomials of any degree, solved in parallel on the GPU. These interpolants are divergence free, periodic on periodic domains, and, with a multiscale kernel, approach Kolmogorov statistics under refinement. A compactly supported kernel, positive semidefinite in any input dimension, makes the operator regression sparse. Its support radius, and hence the sparsity, is fitted to data, and optional affine polynomials in the input capture global trends. Operator Lagrange weights join the two. They cost one sparse solve per new input and act on the training fields before the output reconstruction, so their local coefficients are never formed. We show that, unlike direct evaluation, Lagrange evaluation does not amplify solve errors by the cancellation among spatial coefficients. On ten fluid benchmarks, validation selects sparse matrices nearly everywhere, with densities that mostly decrease as the training set grows, and affine augmentation lowers the error by more than an order of magnitude for forced isotropic turbulence. These sparse models are at least as accurate as our earlier dense method and more accurate than neural operators. Lagrange evaluation is as accurate as direct evaluation with local output reconstructions, and up to nearly three orders of magnitude more accurate with global ones. We also present GPU timings and storage.