K. Nyström

activity

20102026

most citedA unified deep artificial neural network approach to partial differential equations in complex geometries

678 citations · 994 across the 24 of their papers we have counts for

collaborators

36 papers

math.AP2026

Boundary Harnack inequalities for the fractional $p$ -Laplacian in Reifenberg flat domains

Kaj Nyström

We prove a scale-invariant boundary Harnack inequality and Hölder continuity up to the boundary of quotients of positive fractional $p$ -harmonic functions in Reifenberg flat domain…

math.AP2026

Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities

Kaj Nyström

Let $m \geq 1$ , $k \geq 0$ , $n = m + k$ , and write $x = (v, z) \in R^{m} \times R^{k}$ , where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y…

math.AP2026

Solvability of the $\mathrm L^2$ Dirichlet problem for parabolic operators with spatially periodic coefficients

Kaj Nyström

We prove global reverse Hölder estimates in $\mathrm L^2$ for the Poisson kernel of scalar divergence-form parabolic operators in the upper half-space. The coefficient matrix is…

math.AP2026

Higher regularity of regular free boundaries for the Kolmogorov obstacle problem

Kaj Nyström

We establish higher regularity of the regular free boundary for non-negative solutions of the Kolmogorov obstacle problem \[ \mathcal K u=χ_{\{u>0\}}, \qquad \mathcal K=Δ_x+x\cdot\…

math.AP2026

Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains

Kaj Nyström

We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation \[ \mathcal Ku=Δ_xu+x\cdot\nabla_yu-\partial_tu=0 \] in non-characteristic intrinsi…

stat.ML2026

A Commutator Framework for Selective Spectral Alignment in Deep Neural Networks

Kaj Nyström

We develop a finite-width geometric framework describing how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. Incompatibility…

math.AP2023★ 2 cited

A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions

Benny Avelin, Mingyi Hou, Kaj Nyström

In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the doma…

math.AP2023

On fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients

Alireza Ataei, Kaj Nyström

We consider second order degenerate parabolic equations with real, measurable, and time-dependent coefficients. We allow for degenerate ellipticity dictated by a spatial $A_{2}$ -weig…

math.AP2022

On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

Prashanta Garain, Kaj Nyström

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equati…

stat.ML2022

Deep learning, stochastic gradient descent and diffusion maps

Carmina Fjellström, Kaj Nyström

Stochastic gradient descent (SGD) is widely used in deep learning due to its computational efficiency, but a complete understanding of why SGD performs so well remains a major chal…

math.AP2022

Tug-of-war with Kolmogorov

Carmina Fjellström, Kaj Nyström, Matias Vestberg

We introduce a new class of strongly degenerate nonlinear parabolic PDEs $((p - 2) Δ_{\infty, X}^{N} + Δ_{X}) u (X, Y, t) + (m + p) (X \cdot \nabla_{Y} u (X, Y, t) - \partial_{t} u (X, Y, t)) = 0,$ $(X,Y,t)\in\mathb…

math.AP2022

Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

Carmina Fjellström, Kaj Nyström, Yuqiong Wang

The fractional heat operator $(\partial_{t} - Δ_{x})^{s}$ and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous…

stat.ML2021

Solving the Dirichlet problem for the Monge-Ampère equation using neural networks

Kaj Nyström, Matias Vestberg

The Monge-Ampère equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we sol…

math.AP2021

On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

M. Litsgård, K. Nyström

We consider fractional operators of the form $H^{s} = (\partial_{t} - div_{x} (A (x, t) \nabla_{x}))^{s}, (x, t) \in R^{n} \times R,$ where $s \in (0, 1)$ and $A=…

math.CA2021

Parabolic Singular Integrals with Nonhomogeneous Kernels

Simon Bortz, John Hoffman, Steve Hofmann +2

We establish $L^{2}$ boundedness of all "nice" parabolic singular integrals on "Good Parabolic Graphs", aka {\em regular} Lip(1,1/2) graphs. The novelty here is that we include non-h…

math.MG2021

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

Simon Bortz, John Hoffman, Steve Hofmann +2

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows u…

math.AP2020

A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

M. Litsgård, K. Nyström

We consider the operators \[ \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\ \nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X \in Ω$ , $(X,t)…

stat.ML2019

Neural ODEs as the Deep Limit of ResNets with constant weights

Benny Avelin, Kaj Nyström

In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to t…

stat.ML2018★ 160 cited

Data-driven discovery of PDEs in complex datasets

Jens Berg, Kaj Nyström

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to d…

stat.ML2017★ 29 cited

Neural network augmented inverse problems for PDEs

Jens Berg, Kaj Nyström

In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a prior for the coefficient to be estimated…

stat.ML2017★ 678 cited

A unified deep artificial neural network approach to partial differential equations in complex geometries

Jens Berg, Kaj Nyström

In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpro…

math.AP2016

The Dirichlet problem for second order parabolic operators in divergence form

Pascal Auscher, Moritz Egert, Kaj Nyström

We study parabolic operators H = $\partial$ t -- div $λ$ ,x A(x, t) $\nabla$ $λ$ ,x in the parabolic upper half space R n+2 + = {( $λ$ , x, t) : $λ$ > 0}. We assume that the coefficients…

math.CA2016

$L^{2}$ well-posedness of boundary value problems for parabolic systems with measurable coefficients

Pascal Auscher, Moritz Egert, Kaj Nyström

We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal re…

math.CA2015★ 32 cited

The weak- $A_{\infty}$ property of harmonic and $p$ -harmonic measures implies uniform rectifiability

Steve Hofmann, Phi Le, José María Martell +1

Let $E \subset R^{n + 1}$ , $n \geq 2$ , be an Ahlfors-David regular set of dimension $n$ . We show that the weak- $A_{\infty}$ property of harmonic measure, for the open set $Ω:= \m…

Boundary Harnack inequalities for the fractional p-Laplacian in Reifenberg flat domains

Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities

Solvability of the \(\mathrm L^2\) Dirichlet problem for parabolic operators with spatially periodic coefficients

Higher regularity of regular free boundaries for the Kolmogorov obstacle problem

Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains

A Commutator Framework for Selective Spectral Alignment in Deep Neural Networks

A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions

On fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients

On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

Deep learning, stochastic gradient descent and diffusion maps

Tug-of-war with Kolmogorov

Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

Solving the Dirichlet problem for the Monge-Ampère equation using neural networks

On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

Parabolic Singular Integrals with Nonhomogeneous Kernels

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

Neural ODEs as the Deep Limit of ResNets with constant weights

Data-driven discovery of PDEs in complex datasets

Neural network augmented inverse problems for PDEs

A unified deep artificial neural network approach to partial differential equations in complex geometries

The Dirichlet problem for second order parabolic operators in divergence form

L2 well-posedness of boundary value problems for parabolic systems with measurable coefficients

The weak-A∞​ property of harmonic and p-harmonic measures implies uniform rectifiability

Boundary Harnack inequalities for the fractional $p$ -Laplacian in Reifenberg flat domains

$L^{2}$ well-posedness of boundary value problems for parabolic systems with measurable coefficients

The weak- $A_{\infty}$ property of harmonic and $p$ -harmonic measures implies uniform rectifiability