The paper introduces L‑SplineFNO, a memory‑efficient neural operator that combines a Fourier neural operator backbone with a learnable tensor‑product B‑spline transfer to map unstructured meshes to a latent Cartesian grid. By using a local normalised scatter, the method preserves spline properties without geometry‑dependent solves, enabling efficient training on up to 26.6 million nodes and inference on 46 million nodes on a single A100 GPU. L‑SplineFNO achieves strong predictive accuracy across six 3D engineering benchmarks, supports zero‑shot super‑resolution, efficient closed‑form spatial derivatives, and provides rigorous prediction bounds via Bernstein convex hulls and Taylor models on a large tokamak model.
arXiv:2607. 15751v1 Announce Type: new Abstract: This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning.
arXiv:2607. 07718v1 Announce Type: cross Abstract: Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations.
arXiv:2609.36806v1 Announce Type: new Abstract: Modern engineering systems, from automobiles to aircraft, are designed by using precise, continuous parametric computer-aided design (CAD) models. Eval...
arXiv:2601. 20496v2 Announce Type: replace-cross Abstract: Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications.
The paper introduces Spectrally Optimised Neural Discretisations (SpeND), a mesh‑free framework that learns discretisation weights from local stencil geometry on unstructured point clouds. By embedding discrete moment conditions into the network architecture, SpeND guarantees polynomial consistency and allows the weights to be optimised for spectral accuracy over a chosen wavenumber band, using an unsupervised Fourier‑mode loss. The resulting operators are PDE‑agnostic, perform well on Poisson, Burgers, and Navier–Stokes equations, and can reduce wall‑clock time by 3–20× compared to existing mesh‑free methods at the same accuracy.
CANTO is a transformer neural operator that directly maps continuous CAD geometry—represented as sequences of NURBS patches—to physical fields without requiring meshing. It tokenizes control points, knot vectors, and weights to predict continuous surface and volume fields at arbitrary query locations. Evaluated on automotive and aircraft aerodynamics benchmarks, CANTO achieves state‑of‑the‑art accuracy, notably a 19.8% reduction in surface‑pressure error on HiLiftAeroML, and enables gradient‑based inverse design, finding lower‑drag designs on AhmedML.
arXiv:2607. 08025v1 Announce Type: new Abstract: While neural PDE solvers have demonstrated significant potential for accelerating engineering simulations, existing architectures remain constrained by high memory consumption and the single node bottleneck, where the maximum processable mesh resolution is strictly limited by the VRAM of a single compute unit.
arXiv:2607. 06026v1 Announce Type: new Abstract: We present a novel isogeometric deep learning method, termed SplineNet, for the seamless design and analysis of shell structures with complex geometries.
arXiv:2607. 18020v2 Announce Type: replace Abstract: Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators.
The paper introduces MENO, a Memory‑Efficient Neural Operator designed for solving partial differential equations (PDEs). MENO leverages a Manifold Function Encoder to achieve a small memory footprint that does not depend on data resolution, enabling faster training and potential scalability to large models. It accepts PDE inputs of arbitrary form—including different geometric domains and discretizations—allowing cross‑geometry scenarios, and demonstrates strong generalization with superior accuracy on most tested benchmarks.
The paper introduces beignet, a neural field architecture that replaces the random Fourier feature embedding in physics-informed neural networks with a trainable multi‑resolution Fourier feature pyramid. By using Fourier interpolation and spectral derivatives, beignet efficiently computes spatial derivatives and scales accuracy through the feature pyramid rather than the neural network size. Experiments show that beignet achieves more accurate PDE solutions with fewer parameters than existing PINN methods and can reach machine‑precision residuals on the inviscid Burgers blowup problem using the Adam optimizer.
arXiv:2607. 18020v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators.