Related papers: On the structure of measures constrained by linear…
This paper addresses the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs). Contrary to most existing approaches, it considers a framework of structured model learning, where existing,…
Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical…
This paper surveys and discusses recent work adapting partial differential equation (PDE) models to discrete structures.
Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The…
We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In…
We establish various forms of the following certainty principle: a set $S \subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that $S$ is a support of the Fourier transform of a sufficiently singular probability measure…
We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…
We give an essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.
We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and…
We investigate the character of the linear constraints which are needed for Poincar\'e and Korn type inequalities to hold. We especially analyze constraints which depend on restriction on subsets of positive measure and on the trace on a…
Solvable structures are exploited in order to find families of explicit solutions to evolution PDEs admitting suitable differential constraints. The effectiveness of the method is verified on several explicit examples.
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in…
After a brief introduction to the Painlev\'{e} property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlev\'{e} tests. The tests are…
The purpose of the present paper is to show few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is…
Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with…
This paper focuses on defining a measure, appropriate for obtaining optimally sparse solutions to underdetermined systems of linear equations.* The general idea is the extension of metrics in n-dimensional spaces via the Cartesian product…
The main purpose of this paper is to provide combinatorial constraints on the constructability of free and nearly free arrangements of smooth plane conics admitting certain ${\rm ADE}$ singularites.
This expository note aims at illustrating weak convergence of probability measures from a broader view than a previously published paper. Though the results are standard for functional analysts, this approach is rarely known by…
We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines for which dimension conservation does not hold and the projection of the measure is…
The construction and role of symmetries for difference equations are now well known. In this paper, the symmetry analysis of the discrete Painleve equations is considered. We assume that the characteristics depend on $n$ and $u_n$ only and…