Latest Neural Nets Solve World's Hardest Equations Faster Than Ever Before

Abstraction: Neural operators learn mappings between function spaces to solve PDE families

Key points:

• DeepONet (Karniadakis, Brown Univ., 2019): bifurcated branch/trunk architecture learns operators mapping functions to functions; solves entire PDE families from a single training run
• Fourier Neural Operator (FNO, Anandkumar/Caltech, 2020): applies Fourier transform within each layer, reducing expensive convolution to simple multiplication in frequency domain
• FNO solved 30,000 Navier-Stokes simulations in 2.5 seconds total; equivalent traditional numerical solver would take 18 hours
• Unlike prior neural PDE solvers, these generalize across initial/boundary conditions and spatial resolutions without retraining
• 112-page mathematical proof confirmed DeepONet is universal and avoids the curse of dimensionality for many cases
• For phenomena lacking known PDEs (e.g., traffic flow), neural operators may be the only viable modeling approach

Connections: Neural Operators · Partial Differential Equations · Scientific Machine Learning · Transformers

Source: https://www.quantamagazine.org/new-neural-networks-solve-hardest-equations-faster-than-ever-20210419/