Learning PDE Tangent Operators from Stochastic Walks

Gradient-based optimization for PDE-constrained inverse problems and inverse design requires accurate solution sensitivities. These sensitivities can be approximated by differentiating a learned solution surrogate, but accurate solution predictions do not guarantee accurate derivatives. Learning sensitivities directly offers an alternative, although obtaining high-accuracy derivative supervision can be costly. We introduce the Context-Aware Linearity-preserving Tangent Operator (CALTO), which learns reusable tangent operators from sparse, noisy directional observations generated with few stochastic paths. Its nonlinear context encoder modulates both sides of linear kernels, with a bias-free readout. This enforces superposition consistent with shared-path observations, jointly constraining one linear operator per base problem. Across four 2D elliptic benchmarks and one 3D CAD diffusion–reaction benchmark, CALTO achieves 0.71–8.88% context-averaged relative $L_2$ error on held-out in-distribution instances, reducing error by 50.7–74.8% against the strongest learned baseline per benchmark. Comparisons with a nonlinear variant support directional linearity under sparse, noisy supervision; controlled coefficient identification further shows that CALTO tangents improve parameter recovery and reduce final residual ratios by 98.9–99.5% when correcting learned primal surrogates.