The paper introduces a Physics‑Informed Method of Group Data Handling (PI‑GMDH) that builds functional representations of coupled physical fields progressively during the solution process, rather than fixing the structure beforehand. Using tensor‑product Chebyshev functions, the method is applied to the incompressible Navier‑Stokes equations in a two‑dimensional time‑dependent Taylor‑Green benchmark, achieving extremely low validation and test losses with a modest number of active functions. Comparative experiments with degree‑by‑degree, all‑terms PI‑GMDH, and reference PINN/KAN configurations demonstrate that selective progressive construction can yield a favorable balance of accuracy, model size, and computational time for this controlled synthetic benchmark.
Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly mat...
Physics-informed neural PDE solvers typically keep a fixed representation while adapting parameters, which limits performance on heterogeneous problems. The authors propose physics-informed neural plasticity, allowing the representation itself to reshape during training. Their ReCAP method dynamically redistributes capacity via local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging, achieving the lowest relative $L^2$ error on five challenging 3D and 4D benchmarks and reducing error by 10.7%–27.5% compared to the best competing solvers.
arXiv:2608.24049v1 Announce Type: new Abstract: Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors d...
arXiv:2606. 27895v1 Announce Type: cross Abstract: Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented.
arXiv:2503.19081v2 Announce Type: replace Abstract: Scientific foundation models (SciFMs) aim to learn generalizable representations of physical systems governed by partial differential equations (PD...
arXiv:2608. 09483v1 Announce Type: new Abstract: Recently, tensor-based physics-informed neural networks (T-PINNs) have received increasing attention.
The paper introduces GeoDiv KAN, a geometry‑gated, divergence‑free Gaussian radial basis function Kolmogorov Arnold network designed to solve time fractional Navier–Stokes equations on complex domains. By incorporating geometry‑dependent feature gating and a streamfunction formulation, the architecture preserves incompressibility and achieves higher predictive accuracy than existing PINN, PIKAN, RBF KAN, and DF RBFKAN models on square, circular, and star‑shaped benchmarks. GeoDiv KAN is further validated on two data‑driven CFD cases: unsteady flow past a circular cylinder and turbulent RANS flow through a complex U‑bend.
arXiv:2512. 19643v2 Announce Type: replace Abstract: Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
arXiv:2606. 11963v1 Announce Type: new Abstract: Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space.
arXiv:2606. 12337v1 Announce Type: cross Abstract: Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative.
arXiv:2610.32167v1 Announce Type: cross Abstract: Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned rep...