Research map / Learning theory and algorithms / Kernels and regression theory
Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a fast-rate localized generalization analysis instead of the prior slow-rate analysis | Deep Ritz and PINN · Schrödinger equation, Sobolev solution class | generalization error rate: O(1/\sqrt{n}) -> O(1/n) | Helped |
| Use the modified Deep Ritz method rather than the current Deep Ritz method | Deep Ritz · Schrödinger equation, Sobolev solution class | H^1 squared error upper-bound rate: n^{-(2s-2)/(d+2s-2)}\log n -> n^{-(2s-2)/(d+2s-4)} | Helped |
| Use a Fourier basis with modified Deep Ritz | Modified Deep Ritz · Schrödinger equation, Sobolev solution class | H^1 squared error upper-bound rate: n^{-(2s-2)/(d+2s-2)}\log n -> n^{-(2s-2)/(d+2s-4)} | Helped |
| Use a Fourier basis rather than a neural network for PINN | PINN · Schrödinger equation, Sobolev solution class | H^2 squared error upper-bound rate: n^{-(2s-4)/(d+2s-4)}\log n -> n^{-(2s-4)/(d+2s-4)} | Helped |
| Use the proposed PINN upper bound rather than the previous bound | PINN · Schrödinger equation, Sobolev solution class | H^2 squared error upper-bound rate: n^{-(2s-4)/(d+4s-8)}\log n -> n^{-(2s-4)/(d+2s-4)}\log n | Helped |
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