Analysis and Applications

While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples. Based on tools from Information-based complexity, recent work by Grohs and Voigtlaender [Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces, preprint (2021), arXiv:2104.02746] developed a rigorous framework for assessing this so-called ”theory-to-practice gap”. More precisely, in that work it is shown that there exist functions that can be approximated by neural networks with ReLU activation function at an arbitrary rate while requiring an exponentially growing (in the input dimension) number of samples for their numerical computation. This study extends these findings by showing analogous results for the ReQU activation function.

For a family of second-order parabolic systems with periodic, oscillating and time-dependent coefficients, we establish the uniform boundary Hölder and Lipschitz estimates for Neumann problems in $C^1$ and $C^{1,\eta }$ cylinders, respectively, by using the convergence rate method. Moreover, we establish the uniform $W^{1,p}$ estimates for initial-Neumann problems in $C^1$ cylinders for $1<p<\infty$ by using the real-variable method. As a by-product, we also obtain the Gaussian estimates for Neumann function and its derivatives.

In this work, we study the no-flux initial-boundary value problem for the migration-consumption taxis system involving singular density-suppressed motility

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in $H^k(k\ge 2)$ in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established.

We consider the problem of approximating a function in a general nonlinear subset of $L^2$, when only a weighted Monte Carlo estimate of the $L^2$-norm is accessible. The concept of sample complexity, i.e. the number of sample points necessary to achieve a prescribed error with high probability, is of particular interest in this setting. Reasonable worst-case bounds for this quantity exist only for particular model classes, like linear spaces or sets of sparse vectors. However, the existing bounds are very pessimistic for more general sets, like tensor networks or neural networks. Restricting the model class to a neighborhood of the best approximation allows us to derive improved worst-case bounds for the sample complexity. When the considered neighborhood is a manifold with positive local reach, its sample complexity can be estimated through the sample complexities of the tangent and normal spaces and the manifold’s curvature.