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Scientific ML

working pillar 9

The bridge between the two halves of this site: learning that respects, replaces, or accelerates part of a solver.

Pages

Page Covers
Surrogates and reduced-order models POD, DMD, Galerkin projection, hyper-reduction, the \(n\)-width barrier
Physics-informed neural networks Residual losses, loss balancing, spectral bias, where PINNs actually win
Neural operators FNO, DeepONet, amortisation, mode truncation, rollout
Graph networks on meshes Message passing as a learned stencil, locality, rollout stability
Hybrid physics-ML coupling Learned closures, a priori vs a posteriori, differentiable solvers

Choosing between them

You have You want Reach for
Many solves of one parametric family Fast repeated queries Neural operator or ROM
Physics and sparse data, unknown coefficients The coefficients PINN
An unstructured mesh and trajectories A learned simulator Mesh GNN
A working solver with one weak sub-model A better sub-model Hybrid coupling
One solve The answer a conventional solver

That last row is not a joke. For a single forward solve of a standard PDE, a conventional method is faster, more accurate, and comes with convergence theory. Every method on these pages earns its keep through amortisation (many queries), inverse problems (data plus physics), or closure (a gap the physics leaves open).

The theme that runs through every page

All of these methods interpolate over a distribution they were trained on, and none of them slow down when asked something outside it. A solver given an unreasonable input either fails to converge or produces a visibly wrong field. A learned model returns a fast, smooth, confident answer regardless.

That asymmetry is why the discipline here is different from the rest of the site: verification cannot rely on refinement, so it has to rely on knowing and recording the training distribution, testing a posteriori rather than offline, and instrumenting the out-of-distribution case at run time.

Connections