arXiv subjects

Hanfei Zhou

Publications and source records attributed to Hanfei Zhou.

6 recordsLinked to original sources

arXiv2026

Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems

We study minimum-norm interpolation in weighted variation spaces generated by ReLU$^s$ ridge functions on the unit ball, where $s$ is a positive integer. For a target $f$ in the weighted variation space and a node set with sufficiently small fill distance $h$, every minimum-norm interpolant satisfies the direct bound $\|f-\mathscr{I}_{X_h}f\|_{L_2(\mathbb{B}^d)}\lesssim_{d,s}h^{(2s+1)/4}\|f\|_{\mathcal{V}_s}$. Under a boundary separation condition on the nodes, we construct a matching lower bound for the worst-case interpolation error. For the inverse problem, we introduce dual certificates and derive an estimate analogous to a Bernstein inequality. Under suitable upper bounds on dictionary size and spectral lower bounds for the atomic Gram matrices, sufficiently fast $L_2$ convergence of minimum-norm interpolants along geometrically decreasing fill distances implies that the target belongs to $\mathcal{V}_s$. As a byproduct, the best $L_2$ approximation error $E_n(f)_{L_2(\mathbb{B}^d)}$ for shallow ReLU$^s$ networks of width at most $n$ satisfies $E_n(f)_{L_2(\mathbb{B}^d)}\lesssim_{d,s}n^{-1/2-(2s+1)/(2d)}\|f\|_{\mathcal{V}_s}$, with the weighted atomic cost of an approximating network bounded by a constant multiple of $\|f\|_{\mathcal{V}_s}$, independently of $n$.

math.NA↗

arXiv2026

Convergence of Least Squares Methods for PDEs on R^d with Shallow ReLU^k Networks

We establish inverse inequalities in Gaussian Sobolev norms for shallow $\mathrm{ReLU}^k$ trial spaces. The estimates identify the sharp regularity threshold and quantify the dependence on parameter separation and bias magnitude. Based on these estimates, we prove convergence of weighted least squares discretizations of linear PDEs on $\mathbb R^d$ using shallow networks with prescribed inner parameters and finitely many residual samples in an expanding ball. Under suitable stability and regularity assumptions, we obtain an error bound of order $h^{k+(d+1)/2-r}δ_\pm^{-(S-r)}$ in the Gaussian Sobolev norm of order $S$, up to logarithmic factors, where $r$ is the PDE order, $h$ is the parameter fill distance, and $δ_\pm$ is the antipodal separation. The convergence results cover deterministic and random sampling and extend to fixed bounded regions by restriction. We construct parameter sets with $δ_\pm\asymp h$, yielding a convergence rate in terms of the network width that matches a lower bound up to logarithmic factors. Numerical experiments support the theoretical results and provide practical guidance on balancing accuracy and stability.

math.NA↗

arXiv2026

Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds

Physics-informed neural networks (PINNs) provide a mesh-free approach to solving high-dimensional PDEs on complex geometries, but their theoretical foundations on manifolds remain limited. Moreover, conventional PINN analyses typically rely on solution smoothness, while PINNs may perform poorly for low-regularity solutions arising from nonlinear hyperbolic equations. In this paper, we develop a weak PINN (wPINN) framework for approximating entropy solutions of geometry-compatible hyperbolic conservation laws on Riemannian manifolds $\mathcal{M}^d$. Building on the well-posedness theory, we establish a localized $L_1$-stability estimate that converts localized entropy residuals into terminal error bounds and leads to a rigorous convergence analysis of the proposed method. We then derive approximation guarantees for time-dependent entropy solutions on manifolds, revealing how approximation errors accumulate over long time horizons. For the quadrature error, we develop a problem-adapted localization complexity analysis and show that, for a fixed adversarial test-network architecture, the solution-network contribution achieves the fast rate $\mathrm{VC}_{\mathcal F}/n$, up to logarithmic factors. The resulting algebraic network-complexity exponent depends only on the intrinsic dimension $d$, rather than the ambient dimension. For fixed localization scales, and up to logarithmic factors and the localization bias, the solution-network statistical exponent matches the corresponding minimax exponent in $d$-dimensional Euclidean Sobolev approximation. Numerical experiments illustrate that the proposed wPINN framework accurately approximates entropy solutions on manifold geometries.

math.NA↗

arXiv2026

A Quantitative Approximation Framework for Flow Distillation in Diffusion Models

We develop a quantitative framework for diffusion distillation by viewing few step sampling as approximation through compositions of learned flow maps. For trajectory distillation of the probability flow ODE, we show that low noise multimodal regimes separate score approximability from dynamical stability: the score remains efficiently approximable, while small local errors may be strongly amplified by stiff flow dynamics. In a Gaussian mixture Ornstein--Uhlenbeck model, we prove time uniform $L^p(p_t)$ score approximation by ReLU and ReQU networks with explicit polylogarithmic complexity, and derive a computable Lipschitz bound $L(t)$ for the flow velocity. The stability factor $\exp\bigl(\int_s^t L(u)\mathrm du\bigr)$ can grow exponentially as noise decreases and mixture separation increases. Comparing this certificate with a certified local Lipschitz budget for one step students identifies regimes of direct distillation difficulty, without implying an approximation lower bound. We also show that deep residual compositions control global transport error through propagated local errors, and that equalizing cumulative stability yields an optimal nonuniform segmentation. With eight segments, this grid reduces final mean relative MSE by up to $51.9\%$ versus uniform grids.

stat.ML↗

arXiv2026

Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

This paper develops convolutional neural network (CNN) methods for simultaneous Sobolev approximation and elliptic boundary value problems on compact Riemannian manifolds. We prove approximation estimates for single- and multichannel CNNs, with rates governed by the intrinsic dimension and the smoothness gap. Motivated by elliptic stability, we propose a physics-informed CNN framework with a spectral boundary loss. The boundary residual is expanded in boundary Laplace--Beltrami eigenmodes and penalized by Sobolev trace weights, matching the natural $\mathcal H^{2s-1/2}(\partial\mathcal M^d)$ trace norm for $2s$-order elliptic problems. This avoids smooth auxiliary constructions for exact boundary enforcement and singular Sobolev--Slobodeckij double integrals, while allowing FFT-based or precomputed spectral implementations. We also derive an error decomposition separating approximation, generalization, and spectral truncation errors, showing that the proposed loss is aligned with localized fast-rate generalization analysis. Numerical experiments on the upper hemisphere and upper half-torus demonstrate improved accuracy, convergence, and stability over standard PINNs, with one to two orders of magnitude gains for high-frequency boundary data.

cs.LG↗

arXiv2025

Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation

A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential operators. This paper establishes the first simultaneous approximation theory for deep neural networks on manifolds. We prove that a constant-depth $\mathrm{ReLU}^{k-1}$ network with bounded weights--a property that plays a crucial role in controlling generalization error--can approximate any function in the Sobolev space $\mathcal{W}_p^{k}(\mathcal{M}^d)$ to an error of $\varepsilon$ in the $\mathcal{W}_p^{s}(\mathcal{M}^d)$ norm, for $k\geq 3$ and $s<k$, using $\mathcal{O}(\varepsilon^{-d/(k-s)})$ nonzero parameters, a rate that overcomes the curse of dimensionality by depending only on the intrinsic dimension $d$. These results readily extend to functions in Hölder-Zygmund spaces. We complement this result with a matching lower bound, proving our construction is nearly optimal by showing the required number of parameters matches up to a logarithmic factor. Our proof of the lower bound introduces novel estimates for the Vapnik-Chervonenkis dimension and pseudo-dimension of the network's high-order derivative classes. These complexity bounds provide a theoretical cornerstone for learning PDEs on manifolds involving derivatives. Our analysis reveals that the network architecture leverages a sparse structure to efficiently exploit the manifold's low-dimensional geometry. Finally, we corroborate our theoretical findings with numerical experiments.

math.NA↗