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:2608. 12879v1 Announce Type: new Abstract: Fractional partial differential equations describe nonlocal dynamics, but discovering them from noisy data is difficult because fractional differentiation amplifies high-frequency measurement noise and the derivative orders are unknown.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
arXiv:2606. 17460v1 Announce Type: new Abstract: Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows.
arXiv:2607. 29038v1 Announce Type: new Abstract: Fractional scientific machine learning requires numerical operators that can be differentiated, batched, accelerated, and composed with neural networks.
arXiv:2505. 11766v4 Announce Type: replace Abstract: Neural Operators (NOs) are powerful architectures for learning mappings between function spaces.
The paper introduces KAFlow, a low‑rank Kolmogorov‑Arnold flow that learns the generator of physical operators as a compact symbol composed of univariate Kolmogorov‑Arnold functions. By fixing the gauge of these functions, integrating dynamics with exact exponential integrators, and automatically distilling closed‑form expressions, KAFlow achieves markedly lower out‑of‑distribution loss and higher precision on synthetic benchmarks, long‑horizon quantum dynamics, and variable‑order fractional operators. The method recovers operator coefficients to within 10⁻⁴ and discovers complex dispersion relations that sparse regression cannot capture.
The paper introduces Spectral-like Neural Discretisation (SpeND), a mesh‑free method that learns stencil weights via a neural network to approximate the modal response of a spectral operator across a specified band of wavenumbers. By projecting the network output onto the space of polynomial‑consistent weights, SpeND ensures exact consistency while minimizing dispersion and dissipation errors in a self‑supervised, physics‑agnostic manner. Experiments on disordered 2‑D node sets demonstrate that the learned fourth‑order operator matches the exact spectral response over a wider band than traditional LABFM or structured‑grid finite differences, and retains fourth‑order convergence upon refinement.
The article discusses how automatic differentiation (AD) enables neural networks to compute derivatives of governing equations to machine precision, forming the backbone of physics-informed machine learning. However, the author argues that numerical exactness does not guarantee physical fidelity, as AD lacks awareness of the physical structure a solution must satisfy. The paper highlights that many physical constraints—such as convection directionality, diffusion, dispersion, conservation, thermodynamic consistency, symmetry, symplectic structure, positivity, monotonicity, and boundedness—are overlooked by AD, suggesting a need to rethink how PDE-driven neural surrogates are built.
arXiv:2507.07292v2 Announce Type: replace Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
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 StablePDENet, a physics-informed adversarial training framework that regularizes the residual sensitivity of neural operators to improve stability against input perturbations. The method formulates operator learning as a min–max optimization, where a physics-based projected‑gradient adversary generates perturbations and the outer objective combines the attacked physics loss with a normalized residual‑sensitivity penalty. Experiments on benchmark problems show that StablePDENet outperforms PI‑DeepONet and its adversarial variant in accuracy under adversarial attacks while maintaining competitive performance on clean data, and it also enhances generalization and highlights the distinction between model sensitivity and intrinsic operator ill‑conditioning.