sobolev-space formalization

This work addresses the lack of quantitative characterization of approximation capabilities of neural operators in Sobolev norms, which are crucial for the well-posedness, stability, and generalization of partial differential equations (PDEs). We establish, for the first time, a functional analytic framework for neural operators in Sobolev spaces, rigorously proving that their $H^t$ approximation error can be explicitly controlled by the number of network parameters and deriving a power-law relationship between the error and parameter complexity. Using Fourier Neural Operators (FNOs), an $H^1$-norm loss, and high-fidelity numerical simulations, we achieve an $H^1$ test error as low as $10^{-7}$ (corresponding to a relative error of approximately $10^{-3}$) on the Burgers equation. The experimental results align closely with theoretical predictions, validating the efficacy of the proposed framework.

This work addresses the absence of generalization error theory for multi-input neural operators in Sobolev spaces, particularly when input functions are defined on heterogeneous domains with differing dimensions and regularity. The paper establishes the first unified Sobolev generalization framework by integrating function approximation theory, Sobolev space analysis, and statistical learning theory. It derives complexity-dependent approximation and generalization error bounds of logarithmic-logarithmic over logarithmic type, quantifies the contribution of each input space to the overall error, and reveals the coupling mechanism among input dimensionality, regularity, and Sobolev smoothness order. The resulting theory is applicable to operator learning tasks in PDE solving and scientific computing, accurately characterizing the impact of multi-input interactions on learning performance under balanced settings.

This work investigates the optimal $L^p$-approximation capability of deep ReLU networks under joint constraints on width $W$ and depth $L$, for functions in Sobolev spaces $W^{s,q}([0,1]^d)$ and Besov spaces $B^s_{q,r}([0,1]^d)$. We propose a novel sparse vector encoding scheme based on variable-width–depth architectures, integrating tools from approximation theory and function space embeddings. Under generalized Sobolev embedding conditions, we establish— for the first time—the tight convergence rate $O((WL)^{-2s/d})$, up to logarithmic factors, thereby achieving the theoretical optimum. Our analysis unifies and extends prior bounds derived under either fixed-width or fixed-depth assumptions, yielding the sharpest known characterization of expressive power for deep neural networks in terms of the joint $(W,L)$-scaling.

This work addresses the learning of solution maps for parametrized partial differential equations (PDEs) defined on varying domains. Methodologically, it formulates the solution map as a continuous mapping from a metric space of domain deformations to a Banach space of solutions—bypassing restrictive assumptions of diffeomorphism or continuous deformation. It introduces a dual-domain-to-domain (D2D) and domain-to-solution (D2E) mapping strategy, coupled with linear-preserving neural operators (e.g., MIONet), and establishes rigorous convergence guarantees under the star-shaped domain assumption. Theoretically, this is the first framework to provide provable convergence for learning solution maps of variable-domain PDEs while preserving linearity with respect to source terms for linear PDEs. Experimentally, a single trained model generalizes across a broad class of homeomorphic domains, significantly enhancing prediction robustness and generalization under geometric variations.

This work investigates the generalization behavior of norm-minimizing interpolants in Sobolev spaces under noisy data, revealing a persistent and non-vanishing generalization error—referred to as benign overfitting—even in the large-sample regime. By introducing geometric arguments combined with Sobolev inequalities, the analysis is extended for the first time from Hilbert spaces (corresponding to \( p = 2 \)) to general Sobolev spaces with arbitrary \( p \in [1, \infty) \). The study identifies harmful neighborhoods near training points where interpolation amplifies noise. Under assumptions on label noise and data distribution regularity, it is shown that the generalization error of smoothness-preferring interpolants is, with high probability, bounded below by a positive constant. This underscores the critical role of function space selection in determining generalization performance.

This work addresses the universal approximation problem for nonlinear $k$-times differentiable operators and their derivatives in infinite-dimensional Banach spaces. By leveraging an encoder–decoder architecture—encompassing models such as DeepONets—and integrating Bastiani differentiability, the compact-open topology, and a novel weighted Sobolev space framework, we extend classical universal approximation theorems to the setting of infinite-dimensional operator learning for the first time. We establish the first universal approximation theorem guaranteeing uniform approximation of nonlinear operators and all their derivatives up to order $k$ on compact sets, under a broad class of finite input measures. This result provides a rigorous theoretical foundation for high-order-accuracy operator learning, numerical solution of infinite-dimensional PDEs, and optimization problems constrained in Banach spaces.

This work addresses the computational intractability of the classical $H^{-1}$ norm for weak solutions, which requires taking a supremum over an infinite-dimensional space of test functions. To circumvent this limitation, the authors introduce a stochastic weak solution framework that employs spatially localized random test functions and rigorously establishes the equivalence between the expected square of the resulting stochastic residual and the $H^{-1}$ norm. This equivalence enables a variational formulation that obviates the need for a pre-specified deterministic test function space. Integrating this approach with physics-informed neural networks (PINNs) and L-BFGS optimization, the method achieves relative errors below 1% in merely a few hundred training steps across eight challenging second-order linear elliptic problems, substantially outperforming standard PINNs and demonstrating broad applicability to partial differential equations and operator equations in Hilbert spaces.

This work proposes an operator surrogate modeling framework for partial differential equations (PDEs) and boundary integral equations (BIEs) defined on domains that are diffeomorphic to a reference shape. By leveraging domain pullback and parametric mappings, geometric variations are encoded as parameters, thereby recasting the problem as a parametric PDE. Building upon this formulation, neural and spectral operator models are constructed to approximate the mapping from shape parameters to solution fields. The study establishes, for the first time, unified approximation error bounds with explicit convergence rates for cross-shape generalization tasks. Through a theoretical analysis combining parametric PDE theory, complex analyticity, and principal component-based shape encoding, the approach is shown to guarantee shape-family-uniform error estimates for both elliptic and parabolic PDEs as well as BIEs.

This work investigates the regularity of solutions to elliptic boundary value problems in Barron space and its implications for error estimation in the deep Ritz method. Focusing on harmonic functions with Barron boundary data, it establishes for the first time that such solutions generally lack Lipschitz or $H^2$ regularity, yet can be efficiently approximated by Barron functions with low norm. By integrating elliptic regularity theory, Barron space analysis, and approximation theory for ReLU neural networks, the study proves that on half-spaces and two-dimensional rectangular domains, achieving $\varepsilon$-accuracy requires only a Barron norm scaling like $|\log \varepsilon|$. This result yields explicit priori error bounds for the deep Ritz method in both Lebesgue and Sobolev norms, providing rigorous theoretical support for neural network-based PDE solvers.