Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.
problem Learning elliptic PDEs with noisy data and non-homogeneous boundary conditions.
method Bayesian approach with a Hölder space prior on neural network weights.
result Posterior contracts at near-minimax rate without prior knowledge of solution smoothness.
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine para…
Study moduli spaces of elliptic PDEs using derived C∞-geometry.
problem Representability of moduli spaces of solutions of elliptic PDEs.
method Derived C∞-geometry, stacks of relative jets, nonlinear Fredholm analysis. result Moduli stack of solutions is relatively representable by quasi-smooth derived C∞-schemes. Paper studies deep learning for solving elliptic PDEs, proving optimal bounds and neural scaling laws.
problem Solving elliptic PDEs from random samples using machine learning.
method Deep Ritz Method and Physics-Informed Neural Networks (PINNs) for the Schrödinger equation.
result Proves minimax optimal bounds and neural scaling laws for deep PDE solvers.
Paper analyzes DRM for solving high-dimensional elliptic PDEs with generalization bounds.
problem Analyzing generalization error of neural network methods for high-dimensional PDEs.
method Developed a new solution theory for spectral Barron space and derived generalization error bounds.
result Generalization error bounds are independent of dimension and solutions lie in spectral Barron space.
We show, by modifying Borbély's example, that there are 3-dimen\-sional Cartan-Hadamard manifolds M, with sectional curvatures ≤−1, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
Maximal solution of a PDE shows boundary smoothness for certain domains.
problem Boundary behavior of solutions to a specific PDE.
method Reduction to a nonlinear Fuchsian elliptic PDE.
result Hyperbolic radius is smooth up to the boundary.
Interdisciplinary study linking potential theory and elliptic PDEs.
problem Understanding solutions to nonlinear elliptic PDEs.
method Combining geometric and potential theory approaches.
result Validity of comparison principle and existence/uniqueness of solutions.
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from th…
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 studies viscosity solutions in unique Martinet spaces.
problem Properties of viscosity solutions in Martinet spaces.
method Established properties and proved uniqueness of solutions.
result Uniqueness of viscosity solutions in Martinet spaces.
We introduce an elliptic regularization of the PDE system representing the isometric immersion of a surface in R3. The regularization is geometric, and has a natural variational interpretation.
Study proves unique compactification of hyperbolic space.
problem Proving uniqueness of compactification of hyperbolic space.
method Analyzing one-parameter family of elliptic PDEs on hyperbolic space.
result Euclidean half-plane is the only compactification of hyperbolic space.
In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…
The paper shows how neural networks can approximate PDEs with polynomial scaling in dimension.
problem Understanding the complexity of approximating PDE solutions with neural networks.
method Developed a proof technique to simulate gradient descent using neural networks.
result Neural network parameters scale polynomially with input dimension for approximating PDE solutions.
We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth soluti…
We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqu…
We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with g0 the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, D…
Proves uniqueness of capillary disks in 3D domains.
problem Proving uniqueness of capillary disks in three-dimensional domains modeled by elliptic PDEs.
method Using elliptic PDEs and properties of surfaces in 3D domains, generalizing Nitsche's and Hopf's theorems.
result Generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball.
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
New method of symmetrization applied to PDEs on spheres.
problem Estimating solutions of quasilinear elliptic PDEs with singular data.
method Symmetrization method applied to mappings on the sphere, using conformal transformations.
result Estimates solutions of PDEs with Dirac measures on spheres.
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between p-parabolicity and the comparison principle for the p-Laplace equation. We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…
Generalizes Candel's theorem on curvature of laminated surfaces.
problem Finding curvature of laminated surfaces given certain conditions.
method Proves a generalized theorem using elliptic PDEs and Cheeger-Gromov topology.
result Unique laminated metric exists for given curvature function.
Critical points of scale-invariant curvature energies in 4D are analytic.
problem Analyzing critical points of curvature energies in 4D manifolds.
method Applying Noether's theorem to identify conservation laws and lower order elliptic system of PDEs, then using integrability by compensation and interpolation theory.
result Critical points of scale-invariant curvature energies in 4D are analytic.
The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.
problem Existence and uniqueness of radial solutions to a Weingarten equation in a disk.
method Analyzes the linear Weingarten equation in a disk of small radius, considering elliptic, hyperbolic, and parabolic cases.
result Proves existence and uniqueness of radial solutions in the elliptic case, and no solutions in the hyperbolic case.
The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…
By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are degenerate elliptic and solutions have much lower regularity. Still, one may hope …
A new EnKF method for elliptic PDEs reduces dimensionality for accurate state estimation.
problem Elliptic PDEs in fluid flows make traditional EnKF regularization ineffective.
method Low-rank factorization of the Kalman gain based on the Jacobian spectrum.
result Inference can be performed in a low-dimensional subspace of the state space.
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
PD-PINNs accelerate PINN training by incorporating task-specific dictionaries.
problem Training PINNs is slow and lacks theoretical error bounds.
method Integrates task-dependent dictionaries into PINNs to enhance convergence.
result PD-PINNs achieve faster convergence and bounded prediction errors.
New theory proves representability of PDE solutions without complex machinery.
problem Proving representability of PDE solutions using traditional methods is difficult.
method Developed a new model of derived differential geometry using C∞-bornological rings. result Representability of derived moduli stacks of PDE solutions naturally follows from an Artin-Lurie style theorem.
Researchers create solutions for naked singularities in Einstein vacuum equations.
problem Constructing solutions for the interior region of naked singularities in Einstein vacuum equations.
method Novel self-similarity and study of mixed degenerate elliptic-hyperbolic PDE's.
result Gluing together interior and exterior solutions produces a naked singularity.
Study on deformations of Spin(7)-structures on manifolds.
problem Analyzing deformations of Spin(7)-structures on asymptotically conical manifolds.
method Examined the moduli space of torsion-free, asymptotically conical Spin(7)-structures, showing it is an orbifold for generic decay rates.
result Found that the classical Bryant-Salamon metric on positive spinors on S4 has no continuous deformations as an AC Spin(7)-metric. New neural network approach solves Poisson equations efficiently.
problem Approximating solutions to Poisson equations with Dirichlet boundary conditions.
method Using shallow ReLUα-networks to solve Laplace operator equations. result Neural networks can approximate solutions to the Laplace operator with Dirichlet boundary conditions efficiently.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
New theorems prove uniqueness of solutions to geometric PDEs.
problem Proving uniqueness of solutions to geometric PDEs.
method Analyzing nonlinear elliptic PDEs of divergence form.
result Proved several Moser-Bernstein type theorems.
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. Th…
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
PILNO uses neural operators to solve PDEs efficiently on point clouds.
problem Solving partial differential equations (PDEs) on point cloud data efficiently.
method Physics-informed low-rank neural operator framework combining low-rank kernel approximations and an encoder-decoder architecture.
result PILNO efficiently approximates solution operators of PDEs on point cloud data, satisfying PDE constraints and boundary conditions.
Error estimates for nonlinear PDEs using kernel/GP methods.
problem Error analysis of kernel/GP methods for nonlinear and parametric PDEs.
method Sobolev space error estimates based on minimizing norm property of the solution.
result Dimension-benign convergence rates for smooth solutions.
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Study timelike meridian surfaces in Minkowski 4-space with specific properties.
problem Characterize timelike meridian surfaces with special properties in Minkowski 4-space.
method Analyze different classes of timelike meridian surfaces with constant Gauss curvature, mean curvature, and parallel normalized mean curvature vector field.
result Explicit solutions to PDEs describing timelike surfaces with parallel normalized mean curvature vector field.
Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.
problem Solving partial differential equations on complex geometries with physics constraints.
method Combines admissible NURBS parametrizations and PINN solver for arbitrary geometries.
result High convergence rate and accuracy for most PDEs using Deep NURBS.
Paper develops techniques to solve complex PDEs involving higher cohomology forms.
problem Develop PDE techniques to study real (p, p) forms on Hermitian manifolds.
method Parabolic approach to establish existence of classical solutions.
result Existence of classical solutions for a large class of fully nonlinear equations.