Researchers propose PG-KINN, a physics-informed neural network that combines Kolmogorov-Arnold Networks (KANs) as the trial space with an independent piecewise-polynomial test space under a Petrov-Galerkin weak formulation. The approach uses integration by parts and Gauss-Legendre quadrature to reduce differentiation order and improve conditioning, enabling application to non-self-adjoint, nonlinear, and inverse problems where energy-form methods fail. On benchmarks including crack singularities, hyperelasticity, and inverse parameter identification, PG-KINN outperforms both MLP-based physics-informed networks and KAN-based strong/energy-form alternatives (PIKAN).
Researchers propose P-K-GCN, a framework combining graph convolutional networks, Koopman operator theory, and physics-informed loss functions for spatiotemporal super-resolution on irregular geometries. The method linearizes nonlinear dynamics in a latent space and enforces physical constraints to improve reconstruction fidelity. Theoretical analysis claims guaranteed error reduction via Rademacher complexity bounds. The framework is evaluated on reconstructing high-resolution cardiac electrodynamics from sparse 3D heart geometry measurements.
Researchers introduce a backward Kolmogorov equation framework that reformulates diffusion policy training as a deterministic boundary-value PDE problem in Cameron-Martin space, replacing stochastic score matching. The approach uses a precision-weighted Cameron-Martin loss and a Kolmogorov residual as an inference-time failure detector, yielding convergence guarantees tied to kernel effective rank rather than action dimension. Validation on the PushT manipulation benchmark shows 17% improvement in episode reward and 67.6% reduction in inter-step drift; a 6-station manufacturing scheduling task shows 28.4% lower RMSE than LSTM baselines and 96% reduction in deadlock events via Hamilton-Jacobi reachability certification.
A new arXiv preprint introduces KGRL (Knowledge- and Gradient-Guided Reinforcement Learning), a neuro-symbolic algorithm for Parametrized Action Markov Decision Processes (PAMDPs) where each decision involves both a symbolic action and continuous numerical parameters. KGRL uses a Datalog knowledge base to prune infeasible actions and constrain parameter spaces, then applies gradient-based refinement to estimate optimal parameters during training and deployment. The approach improves sample efficiency and episodic return over state-of-the-art PAMDP baselines, and provides local procedural explanations for its decisions.
A new arXiv preprint derives the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, showing that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under anisotropic noise covariance. The authors convert this identity into a training objective called Exact Posterior Score (EPS) that preserves the input/output structure of standard diffusion pretraining, enabling training from scratch or fine-tuning from a pretrained denoiser. EPS is evaluated on five linear inverse problems across FFHQ and ImageNet, outperforming both training-free and training-based baselines while requiring roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.
This arxiv survey reviews four decades of universal approximation theory for feedforward neural networks, covering classical density results for single-hidden-layer networks and quantitative bounds relating approximation error to network size and target function smoothness. It gives particular emphasis to depth-width trade-offs and the parameter efficiency advantages of deeper architectures for structured function classes. The survey also covers recent theoretical developments on Kolmogorov-Arnold Networks (KANs) as an alternative architectural paradigm with emerging approximation-theoretic analysis.
COGENT is a new architecture combining graph neural networks with Neural Ordinary Differential Equations for continuous-time physical forecasting on irregular geospatial meshes. The model encodes historical system states and forcings into latent dynamics that can be queried at arbitrary future times, avoiding the error accumulation of autoregressive rollout. Evaluated on ice-sheet simulations from the Ice-sheet and Sea-level System Model, COGENT shows improved long-range stability over autoregressive graph baselines. The work introduces training stabilization strategies including rollout-horizon sampling and progressive scheduling.
A new arXiv preprint introduces Error-Conditioned Neural Solvers (ENS), a method for neural PDE surrogates that passes the PDE residual field directly as input to the network at each iteration, enabling iterative self-correction rather than gradient-based residual minimization. The authors demonstrate theoretically and empirically that residual minimization is an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining failures of prior hybrid methods. ENS achieves up to 10× accuracy gains on turbulent Kolmogorov flow across four PDE families, with lower compute cost than hybrid approaches and generalization under distribution shift including zero-shot cross-equation transfer.
A new arXiv preprint introduces Deep Gaussian Processes (DGPs) defined over directed acyclic graphs (DAGs), providing a principled probabilistic framework for compositional function estimation in causal models, multi-fidelity simulations, and gene-regulatory networks. The authors derive theoretical results on prior-collapse behavior and information preservation as a function of graph topology, and propose a structured variational approximation that respects graph dependencies and collider explaining-away effects. Empirical validation covers a protein signalling network and a multi-fidelity heavy-ion collision emulation task, claiming state-of-the-art performance. The work is a methodological contribution at the intersection of probabilistic ML and structured graphical models.