arXiv:2608. 14619v1 Announce Type: new Abstract: This work proposes a new interpretable neural operator framework, termed the Physics Informed Kernel Function Neural Operator (PIKFNO), which explicitly incorporates physics informed kernel functions derived from governing equations into the neural operator architecture.
The paper introduces a neural operator architecture that inherently satisfies homogeneous Dirichlet boundary conditions by constraining each layer’s output to lie within the span of selected Dirichlet eigenfunctions of the Laplacian. This design works for any bounded domain with a Lipschitz boundary and any discretization, avoiding the restrictions of previous methods. The authors prove universal approximation for their architecture and demonstrate its effectiveness on Darcy flow and Helmholtz equation problems.
The paper introduces a linearized Physics-Informed Neural Network (lPINN), a reduced‑order neural basis approach for solving forward and inverse differential equations. In an offline phase, lPINN learns continuous, differentiable neural basis functions from numerical solutions, which are then frozen for new problem instances; the online solution is obtained by minimizing the governing‑equation residual with additional constraints. Experiments on advection‑diffusion, Burgers', and nonlinear pendulum equations show that lPINN achieves lower solution and parameter errors than vanilla PINNs while reducing online inference times by up to three orders of magnitude, and its continuous representation generalizes to finer meshes without retraining.
arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
arXiv:2606. 06164v1 Announce Type: new Abstract: Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data.
arXiv:2510. 10350v3 Announce Type: replace-cross Abstract: Operator learning provides a data-driven approach to approximating solution operators of partial differential equations, but its effectiveness depends strongly on how input and output functions are represented.
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.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
arXiv:2609. 38023v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) embed governing equations into deep learning, but enforce them only through loss residuals, leaving highly oscillatory wave behavior to be discovered by optimization.
Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning.
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.
arXiv:2605. 00760v2 Announce Type: replace Abstract: This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network, the DeepONet framework.