A new arXiv preprint establishes global convergence guarantees for the Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs) when trained via gradient descent to minimize PDE residuals. The result applies to a class of semi-linear PDEs (nonlinear in the solution and its first derivative), resolving a longstanding open question about whether gradient descent can get stuck at non-solution local minima. This provides formal mathematical foundations for two of the most widely used neural PDE solvers in scientific machine learning.
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).
This paper establishes the first global convergence theory for Wasserstein Policy Gradient (WPG), a continuous-control RL optimization method that uses optimal-transport geometry over action distributions. The authors show that the Bellman recursion structure of entropy-regularized RL induces a Polyak–Łojasiewicz (PL) geometry that substitutes for classical convexity, enabling global convergence analysis. Key technical contributions include a statewise KL representation of the soft Bellman residual, a Bellman resolvent identity linking value improvement to relative Fisher information, and a uniform log-Sobolev inequality for the evolving Gibbs policy family. The result yields geometric contraction up to discretization bias, providing theoretical grounding for WPG in continuous-action settings.
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 derives Rademacher complexity-based generalization bounds for learning to predict intermediate iterates of projected gradient descent solvers applied to box-constrained quadratic programs. The authors propose a k-neighborhood data collection strategy that augments converged-solution datasets with intermediate solver states, increasing training data without additional solver runs. The work connects to GLENS, a data-efficient global search method, and frames the approach within the Dynamic Data-Driven Application Systems (DDDAS) paradigm for tightening data-model-optimization loops.
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.
A preprint from arXiv frames as an open problem whether AdamW, the dominant optimizer for LLM pretraining, can achieve rigorous convergence guarantees under heavy-tailed stochastic gradient noise. The authors note that sign-based optimizers like Lion and Muon already have sharp heavy-tailed convergence rates, while AdamW's second-moment accumulator may create a fundamental obstruction by hiding large gradients. The paper proves a positive weighted-metric benchmark and introduces a corridor lower-bound mechanism to characterize the potential failure mode.
A new arXiv preprint introduces negative-shifted gradient descent as a method for mixed-sign spectral regularization in overparameterized linear regression, escaping structural limitations of the negative-ridge endpoint. The authors identify a Marchenko-Pastur barrier in a Gaussian spike-plus-flat model and prove that early-stopped paths improve on all admissible endpoints by a polynomial factor in risk under explicit conditions. The main theorem handles general high-effective-rank tails and recovers all head scales simultaneously, with technical control via localized Duhamel integrals and a finite-grid hold-out inequality for validation-selected algorithms.
A new arXiv preprint proves that gradient equilibrium (GEQ), a recently introduced online optimization framework generalizing first-order stationarity, is algorithmically equivalent to Blackwell approachability. The equivalence implies GEQ is also equivalent to regret minimization and calibration, resolving an open question about GEQ's place in the online learning landscape. The reductions are efficient and allow transfer of refined guarantees like optimism and strong adaptivity from regret minimization to GEQ, with applications including online conformal prediction.