Fig. 3: 1D Poisson equation.
From: One-shot learning for solution operators of partial differential equations
a The training data includes a random \({f}_{{\mathcal{T}}}\) generated from GRF and the corresponding solution \({u}_{{\mathcal{T}}}\). b The convergence of L2 relative errors of different approaches for various test cases. The shaded regions represent one standard deviation of the L2 relative errors from 100 test cases. c Testing examples of random f = f0 + Δf with Δf sampled from GRF of σtest = 0.02, 0.05, 0.10, 0.15 and ltest = 0.1. d Prediction example of different approaches for various test cases. e Prediction errors (the geometric mean and standard deviation) when training by frandom with amplitude σtrain and testing on functions with amplitude σtest. The length scales ltrain = 0.01 and ltest = 0.10 are fixed. f Prediction errors (the geometric mean and standard deviation) when training by frandom with different length scale ltrain and amplitude σtrain = 0.10, and testing on functions with amplitude σtest = 0.10 and ltest = 0.10.
