Fast Learning Rates for Physics-Informed Kernel Methods
Theoretical analysis proves finite-sample learning rates for physics-informed kernel estimators, showing differential observations can improve rates from n^-1/4 to n^-1/2.
The paper analyzes a physics-informed kernel estimator combining n value observations and m differential observations for a linear differential operator D, asking how much differential information improves prediction. The authors prove finite-sample bounds, supported by simulations, revealing a two-regime structure: when m is limited the rate depends jointly on n and m, and when m exceeds a problem-dependent threshold the rate saturates to the oracle rate. Examples in Sobolev spaces, including partial Laplacian constraints on the torus and gradient observations on bounded domains, illustrate improvements from the nonparametric n^-1/4 rate to the parametric n^-1/2 rate, plus physically consistent rates in a stronger norm.
- Proves two-regime finite-sample bounds for physics-informed kernel estimation
- Sufficient differential observations yield oracle rates matching perfect constraints
- Rates range from nonparametric n^-1/4 to parametric n^-1/2
- Covers Sobolev-space examples with partial Laplacian and gradient constraints
From the original article
1 min read at arxiv.orgIn physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+ξ_j$ or by a known physical constraint $Du^*=v$.
By Luc Brogat-Motte · arxiv.org
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