arXiv:2609. 23529v1 Announce Type: new Abstract: Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment.
arXiv:2512. 09165v2 Announce Type: replace Abstract: Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs).
The paper introduces WaSP, a Wasserstein Data Synthesis Pipeline that trains neural operators on manufactured input‑solution pairs while progressively reducing distribution shift toward the target input distribution. By iteratively generating new pairs through a learned surrogate and the exact PDE operator, WaSP achieves a stepwise error‑propagation bound and empirically reduces solution error by 3.9× to 9.5× on PDE benchmarks such as Allen‑Cahn, Kuramoto‑Sivashinsky, and Ginzburg‑Landau.
arXiv:2605. 08170v2 Announce Type: replace Abstract: Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces.
arXiv:2609.36709v1 Announce Type: cross Abstract: Out-of-distribution (OOD) generalization is a central challenge in scientific machine learning. We study regression problems in which the test distri...
arXiv:2609.36047v1 Announce Type: cross Abstract: Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional....
arXiv:2606. 02427v1 Announce Type: cross Abstract: Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure.
arXiv:2602. 09530v2 Announce Type: replace-cross Abstract: We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization.
arXiv:2609. 18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$.
The paper introduces a kernel‑based operator learning method that preserves key physical properties—such as incompressibility, periodicity, and turbulence—of the incompressible Navier–Stokes equations. By mapping input functions to expansion coefficients in a property‑preserving kernel basis, the method guarantees that predicted velocity fields analytically maintain these properties. The authors provide theoretical convergence guarantees, develop efficient computational techniques for large‑scale training, and demonstrate significant accuracy and speed improvements over neural operators on 2D and 3D flow benchmarks.
The paper investigates how many linear samples are needed to learn Lipschitz operators under Gaussian measures. It establishes both lower and upper bounds on the Hermite polynomial approximation error and shows that the minimal worst‑case error cannot converge algebraically with the number of samples. However, if the covariance operator of the Gaussian measure decays rapidly, convergence rates arbitrarily close to any algebraic rate can be achieved.
arXiv:2606. 30230v1 Announce Type: cross Abstract: Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training.
arXiv:2608. 11019v1 Announce Type: new Abstract: Modeling spatiotemporal dynamical systems governed by partial differential equations (PDEs) poses two major challenges: it either requires expensive physics-based simulators that entail iterative numerical solving at high computational cost, or it depends on abundant training data, yet purely data-driven models often generalize poorly to downstream dynamic operating conditions.