Develops a new approach to describe gauge theories with background fields using presymplectic structures.
problem Describing gauge theories with background fields using presymplectic structures.
method Extension of the presymplectic BV-AKSZ approach to include background fields.
result Gauge theories with background fields correspond to presymplectic gauge PDEs over gauge PDEs describing background fields.
Solves the gauge problem in diffeomorphisms for non-compact spaces.
problem Recognizing metrics in different coordinates, especially in non-compact spaces.
method Solves a nonlinear system of PDEs to produce a diffeomorphism that fixes an appropriate gauge.
result Shows optimal bounds for the displacement function of the diffeomorphism.
Solves the gauge problem for Ricci flow cylinders, proving strong rigidity.
problem Recognizing metrics in different coordinates and diffeomorphisms.
method Solves a nonlinear system of PDEs to produce a diffeomorphism fixing a gauge.
result Strong rigidity of cylinders in Ricci flow, proving all tangent flows are cylinders.
Fundamental solutions found for PDEs in Finsler geometry.
problem Solving nonlinear PDEs in Finsler geometry.
method Introduced a non-isotropic Minkowski gauge and computed fundamental solutions.
result Explicit fundamental solutions computed for the PDEs.
A new principle minimizes residual and introduces momentum to improve PDE solution dynamics.
problem Ill-conditioning in Dirac-Frenkel residual minimization leads to non-unique parameter dynamics.
method Introduces a history variable (momentum) to select better-conditioned parameter velocities, preserving residual minimization while promoting smooth parameter evolutions.
result The approach leads to increased robustness in singular and near-singular PDE solution regimes.
We generalise to the Z2-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in …
Survey on recent developments in isometric immersions using PDE techniques.
problem Analyzing isometric immersions with low Sobolev regularity.
method Compensated compactness and Coulomb-Uhlenbeck gauges.
result Weak continuity and stability of Gauss-Codazzi-Ricci equations.
Study boundary structure of gauge fields on AdS spaces.
problem Understanding boundary conditions of gauge fields on asymptotically AdS backgrounds.
method Employing gauge PDE approach to incorporate boundary-defining function.
result Construction of efficient boundary calculus for gauge fields on AdS backgrounds.
Extends RDS filtering to position-orientation space for better image processing.
problem Enhancing and inpainting images with crossing structures.
method Created a version of RDS filtering using gauge frames, studying generalised diffusion.
result RDS filtering on position-orientation space improves denoising and inpainting of crossing structures.
Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.
problem Understanding the geometry of differential equations and their solutions.
method Develops a spectral sequence related to internal Lagrangians and investigates connections to presymplectic structures.
result Interprets a term in Vinogradov's spectral sequence for gauge theories.
A framework for reducing PDEs by symmetry, preserving key structures.
problem Reducing PDEs while preserving geometric structures and symmetries.
method Systematic calculation of reduced forms for various geometric structures.
result Noether's theorem is inherited in reduced systems, preserving conservation laws.
Constructs a moduli space for PDEs, linking stability to geometric metrics.
problem Moduli space construction for involutive ideal sheaves from PDEs.
method Introduces D-Hilbert and D-Quot functors, defines Spencer stability. result Spencer poly-stability of PDE ideal implies Hermitian-Yang-Mills metric existence.
Researchers study spectral asymmetry using pseudodifferential projections on the massless Dirac operator.
problem Understanding spectral asymmetry for the massless Dirac operator.
method Constructing a negative order pseudodifferential asymmetry operator from spectral projections.
result Computed the principal symbol of the asymmetry operator, accounting for gauge invariance.
By using the geometric concept of PDEs with prescribed curvature representations, we show that the 1+2 dimensional Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up H3(R2)-solutions to the 1+…
This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over ¶1, and the generalized quiver representations) were included. The main goal is the construction of gauge th…
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A moti…
Let (M,g) be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over M is said to be transparent if its parallel transport along the closed geodesics of g is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms …
We give new estimates for a critical elliptic system introduced by Rivière-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and almost harmonic) maps from a Euclidean ball $B_1 \In \R^n$ into Riemannian manifol…
Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.
problem Proving well-posedness for the Bartnik static extension problem near Schwarzschild spheres.
method Introduced a geodesic gauge to formulate governing equations as coupled elliptic and transport equations; used Bochner-measurable functions for transport equations.
result Established local well-posedness for arbitrary Bartnik data near Schwarzschild spheres, including those with small mean curvature.
In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of …
Unconditional proof of Demailly's transcendental Morse inequality for higher cohomology classes using a general gauge-fixing for the Monge-Ampère-type equation.
problem Unconditional proof of Demailly's transcendental Morse inequality for higher cohomology classes
method General gauge-fixing for the (a,b) Monge-Ampère-type equation result Unconditional proof of Demailly's transcendental Morse inequality for higher-degree forms
We study the equation E_fc of flat connections in a fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms f: E\to E_fc, where E and E_fc…
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
Develops derived differential geometry for supermanifolds.
problem Handling non-transverse intersections and singular moduli problems in geometry and physics.
method Extends existing work on derived manifolds to supergeometric and infinite-dimensional contexts.
result Establishes foundational results relating derived differential geometry to differential operators and PDE theory.
Neural-PDE learns PDEs from data using LSTM, outperforming traditional methods.
problem Solving time-dependent PDEs numerically is challenging.
method Bidirectional LSTM encoder to learn governing rules from data.
result Neural-PDE efficiently predicts PDE dynamics with minimal parameters.
Neural Q-learning tackles high-dimensional PDEs.
problem Solving high-dimensional PDEs is computationally challenging.
method Adapting Q-learning from reinforcement learning to solve PDEs.
result The neural network approximator converges to the PDE solution as the network width increases.
Paper introduces Tensor Gauge Flow Models for better data encoding.
problem Lack of expressive flow dynamics in existing Generative Flow Models.
method Incorporates higher-order Tensor Gauge Fields into the Flow Equation.
result Tensor Gauge Flow Models achieve improved generative performance.
Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net …
PDE-DKL combines NNs and GPs for high-dimensional PDE problems.
problem High-dimensional PDE problems with scarce data.
method PDE-constrained Deep Kernel Learning (PDE-DKL) framework.
result High accuracy with reduced data requirements.
L-CNNs preserve gauge symmetry in lattice simulations.
problem Breaking gauge symmetry in neural network models.
method Lattice gauge equivariant convolutional neural networks (L-CNNs).
result L-CNNs represent gauge invariant functions on the lattice.
Solves second-order PDEs using quotients and differential invariants.
problem Solving second-order PDEs with first-order quotients.
method Solve the quotient PDE using differential invariants, then add new constraints to solve the original PDE.
result New method for solving second-order scalar PDEs with infinite-dimensional symmetry algebras.
PRISMA uses PDE residuals for fast, robust, and accurate inference.
problem Slow gradient-based optimization and instability in PDE residual-based methods.
method Integrates PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain.
result Competitive accuracy with significantly lower inference costs and faster speeds.
L-CNNs learn gauge invariant quantities on lattices.
problem Learning gauge invariant quantities on lattices.
method Novel convolutional layer preserving gauge equivariance and forming Wilson loops.
result L-CNNs can approximate any gauge covariant function on the lattice.
Meta-learning base distributions for efficient PDE solutions.
problem Efficiently solving parametric parabolic PDEs across different scenarios.
method Meta-learning base distributions to compute PDE solutions.
result Improves generalization to new parameter regimes.
L-CNNs preserve gauge symmetry in neural networks.
problem Applying machine learning to lattice gauge theory while preserving gauge symmetry.
method L-CNNs use gauge equivariance to construct a gauge equivariant convolutional layer and bilinear layer.
result L-CNNs achieve higher accuracy in non-linear regression tasks compared to non-equivariant CNNs.
We study the problem of finding good gauges for connections in higher gauge theories. We find that, for 2-connections in strict 2-gauge theory and 3-connections in 3-gauge theory, there are local "Coulomb gauges" that are more canonical than in classical gauge theory. In particular, they are essentially unique,…
When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…
Kernel method learns PDEs from noisy data.
problem Discovering and solving PDEs from noisy data.
method Kernel smoothing, regression, and operator learning.
result Competitive performance compared to state-of-the-art algorithms.
Solves a PDE for Landsberg surfaces using new Finsler surface insights.
problem Solving the Landsberg's PDE for Finsler surfaces.
method Reduces the system of non-linear PDEs to a single PDE, the Landsberg's PDE, and solves it.
result Obtains a class of solutions for the Landsberg's PDE.
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential equations.
Develops arithmetic PDE geometry concepts like curvature and cohomology.
problem Creating a geometry framework for arithmetic PDEs.
method Introducing arithmetic analogues of Levi-Civita and Chern connections, then developing curvature and characteristic classes.
result Arithmetic analogues of curvature and characteristic classes have been developed.
Unified framework solves nonlinear PDEs and IPs using Gaussian processes.
problem Solving and identifying parameters in nonlinear PDEs and inverse problems.
method Gaussian process framework approximating solutions as MAP estimators, reducing to finite-dimensional optimization problem.
result Unified method converges in a small number of iterations for various PDEs.
We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a recent work on transferring curved A∞-structures we show that, under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Sp…
We consider dimensional reduction of gauge theories with arbitrary gauge group in a formalism based on equivariant principal bundles. For the classical gauge groups we clarify the relations between equivariant principal bundles and quiver bundles, and show that the reduced quiver gauge theories are all generically buil…
The paper presents a PDE method for xVA incorporation in financial derivatives.
problem Incorporating value adjustments (xVA) in financial derivative pricing.
method Analytical solution of PDEs in the Black-Scholes framework.
result New semi-closed formulas for xVA are derived and compared to Monte-Carlo and numerical methods.
We consider gauged twistor spinors which are supersymmetry generators of supersymmetric and superconformal field theories in curved backgrounds. We show that the spinor bilinears of gauged twistor spinors satify the gauged conformal Killing-Yano equation. We prove that the symmetry operators of the gauged twistor spino…
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.