Discretization-independent operator learning for partial differential equations
Read the original on arXiv Machine Learning →The Flow has not summarised this story yet — read it at arXiv Machine Learning.
The Flow has not summarised this story yet — read it at arXiv Machine Learning.
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
The paper introduces the Frame Kernel Method, a multiscale operator learning approach for surrogate modeling of multiscale partial differential equations. It uses a novel multiscale kernel frame function approximation to cast the learning problem as one of estimating frame coefficients, enabling automatic multiscale decomposition of outputs. The authors provide interpolation proofs, error estimates, and demonstrate that the method outperforms popular neural operators on challenging PDE problems while offering a posteriori multiscale analysis.
arXiv:2607. 12570v1 Announce Type: cross Abstract: Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations.
The paper introduces two multi-stage neural operator learning frameworks—Deep Collocation Neural Operator (DCNO) and Deep Galerkin Neural Operator (DGNO)—for efficiently computing convolution integrals. DCNO is a supervised method that iteratively refines operator approximations by learning residuals from data pairs, while DGNO is an unsupervised approach that uses the weak form of a PDE residual when the operator can be represented by a PDE. Both frameworks build basis operators across multiple training stages, yielding markedly higher accuracy than one-shot learning and achieving near machine‑precision results for convolution problems, with significant efficiency gains for repeated queries or parametric variations.
arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
arXiv:2512.12749v3 Announce Type: replace-cross Abstract: Learning surrogate models for physical systems with latent uncertainty remains challenging in data-scarce regimes: deterministic neural opera...