🤖 AI Summary
This work addresses the challenges of solving high-dimensional, high-order parametric partial differential equations (PDEs), which suffer from prohibitive computational costs and the need for repeated training in existing stochastic methods. The authors propose PRISM, a novel framework that decouples parameterization from high-order spatial differentiation by implicitly mapping physical parameters—via a hypergenerator—into affine modulators acting on a purely spatial latent manifold. This approach preserves unbiased stochastic estimation, incorporates a variance-aware Lipschitz envelope, and provides theoretical guarantees on error bounds and convergence. PRISM enables zero-shot generalization to unseen parameters, drastically reduces memory consumption, scales to problems with up to one hundred thousand dimensions on a single GPU, and supports efficient low-rank adaptation without retraining.
📝 Abstract
Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.