Related papers: Benign overfitting in Fixed Dimension via Physics-…
We establish a unified theoretical framework addressing the stability, consistency, and convergence of neural networks under realistic training conditions, specifically, in the presence of non-IID data, geometric constraints, and embedded…
Establishing a theoretical analysis that explains why deep learning can outperform shallow learning such as kernel methods is one of the biggest issues in the deep learning literature. Towards answering this question, we evaluate excess…
We characterize and remedy a failure mode that may arise from multi-scale dynamics with scale imbalances during training of deep neural networks, such as Physics Informed Neural Networks (PINNs). PINNs are popular machine-learning templates…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$,…
An asymptotic theory is established for linear functionals of the predictive function given by kernel ridge regression, when the reproducing kernel Hilbert space is equivalent to a Sobolev space. The theory covers a wide variety of linear…
Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational…
The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The…
We consider the problem of supervised learning with convex loss functions and propose a new form of iterative regularization based on the subgradient method. Unlike other regularization approaches, in iterative regularization no constraint…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
Kernels are efficient in representing nonlocal dependence and they are widely used to design operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal…
This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for…
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its…
In many real world applications of machine learning, models have to meet certain domain-based requirements that can be expressed as constraints (e.g., safety-critical constraints in autonomous driving systems). Such constraints are often…
Smoothness is a key inductive bias in machine learning and is closely related to generalization. Existing smoothness-inducing methods typically rely either on explicit gradient regularization, which often incurs substantial computational…
This letter presents a high-dimensional analysis of the training dynamics for a single-layer nonlinear contrastive learning model. The empirical distribution of the model weights converges to a deterministic measure governed by a…
We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is…
We study Sobolev regularity results for minimisers of autonomous, convex variational of linear growth which depend on the symmetric gradient rather than the full gradient. This extends the results available in the literature for the…
We establish that the Dirichlet problem for convex linear growth functionals on $BD$, the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $C^{1,\alpha}$-regularity theory as presently available for…
A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal…