The paper presents a theoretical framework for certifying the accuracy of physics‑informed neural networks (PINNs) used to solve partial differential equations. It derives generalization bounds that link the residual loss minimized during training to the actual error in the solution space, showing that if the neural approximation stays within a compact subset, a vanishing residual guarantees convergence to the true solution. Deterministic and probabilistic convergence results are provided, offering explicit error guarantees based on residual, boundary, and initial condition errors.
By Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fern\'andez, Jun Liu
The paper introduces StablePDENet, a physics-informed adversarial training framework that regularizes the residual sensitivity of neural operators to improve stability against input perturbations. The method formulates operator learning as a min–max optimization, where a physics-based projected‑gradient adversary generates perturbations and the outer objective combines the attacked physics loss with a normalized residual‑sensitivity penalty. Experiments on benchmark problems show that StablePDENet outperforms PI‑DeepONet and its adversarial variant in accuracy under adversarial attacks while maintaining competitive performance on clean data, and it also enhances generalization and highlights the distinction between model sensitivity and intrinsic operator ill‑conditioning.
By Chutian Huang, Chang Ma, Kaibo Wang, Yang Xiang
arXiv:2607. 02003v1 Announce Type: cross Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics.
By Matej Benko, Pierre Bousquet, Iwona Chlebicka, B{\l}a\.zej Miasojedow
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
Gen-PINNs introduces a deterministic residual-adversarial framework that combines Generative Adversarial Networks with Physics-Informed Neural Networks to solve partial differential equations. The method addresses limitations of standard PINNs—such as spectral bias, loss imbalance, and optimizer stagnation—by using a generator that learns the PDE solution with dynamically weighted physics-informed losses and discriminators that assess residuals against zero-residual states. Experiments on nonlinear and higher-order PDEs (Burgers, Allen‑Cahn, Kuramoto‑Sivashinsky) show significant gains in accuracy and convergence, especially for sharp-front, stiff, and high-order problems.
By Muhammad M. Akmal, Kamy Sepehrnoori, Michael J. Pyrcz
The paper introduces a two‑step debiased estimation method for PDE‑constrained inverse problems where the PDE solution is approximated by Physics‑Informed Neural Networks (PINNs). By combining neural‑network‑based nonparametric estimation with an influence‑function bias correction, the authors achieve a √{n}-consistent, asymptotically normal estimator without undersmoothing the neural network. The approach is extended to Bayesian inference, yielding a posterior that contracts at the √{n}-rate with asymptotic covariance matching the frequentist estimator, and the analysis also provides near‑minimax rates for estimating nonparametric regression functions and their derivatives in Sobolev spaces.
By Yves Atchade, Debarghya Mukherjee