arXiv:2607. 00320v1 Announce Type: cross Abstract: We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations.
arXiv:2606. 08343v1 Announce Type: new Abstract: We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC (metriplectic) structure of nonequilibrium thermodynamics -- reversible, energy-conserving dynamics and irreversible, entropy-producing dynamics coupled through the degeneracy conditions -- directly in function space.
arXiv:2605. 08170v2 Announce Type: replace Abstract: Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces.
arXiv:2607. 27781v1 Announce Type: cross Abstract: We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms.
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.
arXiv:2503. 18219v2 Announce Type: replace Abstract: This work studies the sampling complexity of learning with ReLU neural networks and neural operators.
arXiv:2606. 28122v1 Announce Type: cross Abstract: Neural operators provide deep neural networks for learning mappings between function spaces.
arXiv:2607. 11974v1 Announce Type: cross Abstract: Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen.
arXiv:2606. 17419v1 Announce Type: new Abstract: We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms.
arXiv:2602. 06842v2 Announce Type: replace-cross Abstract: Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate convergence.
arXiv:2606. 17319v1 Announce Type: cross Abstract: Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube.
arXiv:2607. 15702v1 Announce Type: cross Abstract: We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations.
arXiv:2601. 17090v2 Announce Type: replace-cross Abstract: Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps.