New deep learning methods solve symmetric PDEs efficiently.
problem Solving nonlinear symmetric PDEs in high dimensions.
method Design of PointNet and DeepSet neural networks.
result DeepSet networks provide more accurate solutions and gradients.
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48 results for “symmetric PDEs”
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
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.
In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…
Study on curvature blow-up rates in black hole interiors from gravitational collapse.
problem Understanding curvature blow-up rates in black hole interiors during gravitational collapse.
method Investigation of spherically symmetric Einstein-scalar field spacetimes, focusing on blow-up rates of curvature and mass.
result Kretschmann scalar blows up faster than in Schwarzschild setting, indicating a new blow-up phenomenon.
Exceptionally simple PDEmath.DG
We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except . This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the case. We give a …
We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly type equations into new integrable PDE of the second order with large symmetry pseu…
We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in…
New insights into 3D PDEs via Einstein-Weyl geometry.
problem Understanding second-order PDEs in 3D with Einstein-Weyl conformal structure.
method Analyzing solutions of second-order dispersionless integrable PDEs in 3D, relating them to Einstein-Weyl geometry.
result The covector w can be expressed in terms of the equation for generic second-order PDEs, providing a dispersionless integrability test.
Researchers recover Riemannian manifolds and lower order terms from travel time data.
problem Recovering Riemannian manifolds and lower order terms from travel time data.
method Adaptation of the Boundary Control method to recover lower order terms.
result Complete Riemannian manifolds and lower order terms can be uniquely recovered from a local source to solution map.
Compositional diffusion models simulate coupled PDEs efficiently.
problem Efficiently simulating long-horizon coupled PDE systems.
method Diffusion models trained on decoupled data are composed at inference time.
result Compositional diffusion models recover coupled trajectories with low error.
Study of caustics in Einstein-dust system, showing spacetime singularities and diverging curvature.
problem Understanding caustics and singularities in the Einstein-dust system.
method Established local existence result for spherically symmetric spacetimes containing caustics, constructed from solutions to a PDE problem.
result Obtained spherically symmetric spacetimes with diverging curvature and singular boundary.
We define a large class of integrable nonlinear PDE's, \emph{-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the soluti…
We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the Kähler form on a Kähler manifold. A \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of Kähler metri…
Study analyzes symmetric two-armed Bernoulli bandit problem with zero mean gap.
problem Analyzing symmetric two-armed Bernoulli bandit problem with zero mean gap.
method Associated with a solution of a linear heat equation, compute leading order terms of minmax optimal regret and pseudoregret.
result Explicitly compute leading order terms in three scaling regimes for the gap.
For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in…
Enhanced autoencoders improve ROMs for PDEs by capturing essential properties.
problem Autoencoders struggle to capture essential properties for accurate ROMs.
method Introduced symmetric Convolutional AutoEncoders (CAEs) that preserve manifold properties.
result Symmetric CAEs yield more accurate latent trajectories and robust models.
Paper proves optimal decomposition for matrix fields, reducing convex integration steps.
problem Optimizing decomposition of symmetric matrix fields for convex integration.
method Algebraic geometry and topology applications to prove optimality.
result Optimal decomposition with fewer rank-one terms, improving Hölder regularity.
This paper classifies the symmetry groups of a specific Monge-Ampère equation.
problem Classifying the symmetry groups of a specific Monge-Ampère equation.
method Developed Lie theory for fully nonlinear PDEs in Convex Geometry.
result Completely classified the symmetric groups of the -Minkowski problem.
We study several classes of Riemannian manifolds which are defined by imposing a certain condition on the Ricci tensor. We consider the following cases: Ricci recurrent, Cotton, quasi Einstein and pseudo Ricci symmetric condition. Such conditions can be interpreted as overdetermined PDE systems whose unknowns are the c…
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …
We give a new proof for the local existence of a smooth isometric embedding of a smooth -dimensional Riemannian manifold with nonzero Riemannian curvature tensor into -dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …
The study of the -th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…
The defining equations for Killing vector fields and conformal Killing vector fields are overdetermined systems of PDE. This makes it difficult to solve the systems numerically. We propose an approach which reduces the computation to the solution of a symmetric eigenvalue problem. The eigenvalue problem is then solved …
Study uses G-BSDEs to decompose pricing kernels under robust G-expectation.
problem Long-term decomposition of robust pricing kernels under G-expectation.
method Proposes and analyzes three types of quadratic G-BSDEs to decompose pricing kernels.
result Pricing kernels decomposed into four components: discounting, transitory, symmetric martingale, and volatility uncertainty.
A gluing construction of singular solutions for a fully non-linear equation in conformal geometrymath.DG
Paper studies solutions to a specific equation in conformal geometry with singular sets.
problem Singular solutions to a fully non-linear equation in conformal geometry.
method Uses a classical gluing method adapted to the fully non-linear setting.
result Shows the classical gluing method can be applied to the --Yamabe equation.
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…
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing .
For modelling of various physical processes, geodesic lines and almost geodesic curves serve as a useful tool. Trasformations or mappings between spaces (endowed with a metric or connection) which preserve such curves play an important role in physics, particularly in mechanics, and in geometry as well. Our aim is to c…
Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equationsmath.NA
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.
In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short time…
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.
New invariant connects boundary PDEs and conformal geometry.
problem Global conformal invariants of boundary PDEs.
method Variational considerations and conformal invariants.
result Compact Bach-flat manifolds with umbilic boundary admit Poincaré-Einstein metrics.
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.
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 …
We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions , and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sh…
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.
The study of the -th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called curvature, has produced many fruitful results in conformal geometry in recent years, especially when the dimension of the underlying manifold is 3 or 4. In these studies, the deforming…
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.
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.
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.
PDE-based G-CNNs add geometric symmetries to CNNs without augmentation.
problem Designing CNNs with built-in symmetries like rotation.
method Formulate CNN layers as PDE solvers on homogeneous spaces.
result PDE-G-CNNs achieve better performance with fewer parameters.
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.