This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers.
arXiv:2607. 07637v1 Announce Type: new Abstract: This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation.
arXiv:2601. 07397v2 Announce Type: replace-cross Abstract: In this work, we propose a novel layerwise adaptive construction method for neural network architectures.
arXiv:2605. 09075v2 Announce Type: replace-cross Abstract: Although the Laplace approximation offers a simple route to uncertainty quantification in deep neural networks, its reliance on inverting large Hessian matrices has motivated a range of computationally feasible low-dimensional or sparse approximations.
arXiv:2505. 15497v3 Announce Type: replace Abstract: Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems.
arXiv:2604. 20219v2 Announce Type: replace Abstract: Depth is widely viewed as a central contributor to the success of deep neural networks, whereas standard neural network approximation theory typically provides guarantees only for the final output and leaves the role of intermediate layers largely unclear.
arXiv:2608. 06428v1 Announce Type: new Abstract: Deep Operator Networks (DeepONets; arXiv:1910.
arXiv:2402. 00152v5 Announce Type: replace Abstract: Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question.
The paper proposes treating a neural network’s layers as time steps in a state‑space model, converting Bayesian training into a smoothing problem. By propagating Gaussian moments forward and applying a Rauch–Tung–Striebel backward pass, weight posteriors are updated in closed form without gradient iterations or replay. The authors extend prior work by introducing a cross‑covariance identity that allows full‑covariance propagation through nonlinear activations, enabling more accurate online adaptation in non‑stationary classification, dynamics learning, and vision‑language‑action policy adaptation.
The paper introduces a method for constructing L-Lipschitz deep residual networks (ResNets) using a Linear Matrix Inequality (LMI) framework. By reformulating the ResNet architecture as a pseudo-tridiagonal LMI and applying the Gershgorin circle theorem, the authors derive closed‑form constraints on network parameters that guarantee Lipschitz continuity. The work also presents a compositional framework for handling recursive systems in hierarchical architectures, while noting that the Gershgorin-based approximations can over‑constrain the system, reducing expressive capacity.
arXiv:2601.12971v2 Announce Type: replace Abstract: Physics-informed neural networks (PINNs) can be limited by coordinate representations and conflicting gradients from heterogeneous physical constra...
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.
arXiv:2609.07755v1 Announce Type: new Abstract: Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training d...