The paper proved that every -solution of a given first order PDEs system, regarded on the jet fibre bundle of order one , may be viewed as a "generalized harmonic map", via the least squares variational method. Our ideas are structured in the following way: 1) we find a suitable geometrical structure on …
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48 results for “PDE systems”
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.
New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.
problem Developing integrable systems from tensor field properties.
method Using Frölicher-Nijenhuis brackets to generate bi-differential graded algebras and PDE systems.
result New integrable nonlinear matrix PDEs and systems are derived.
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.
The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal sur…
New method uses Gaussian processes for solving linear PDEs with boundary conditions.
problem Solving linear PDEs with boundary conditions.
method Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs).
result Significant accuracy and resource improvements over existing methods.
This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87 (1997), 265-319]. The constrained variational bi-complex is introduced and used to d…
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.
The aim of this paper is fourfold. Firstly, we introduce and study the f-ultra-harmonic maps. Secondly, we recall the geometric dynamics generated by a first order normal PDE system and we give original results regarding the geometric dynamics generated by other first order PDE systems. Thirdly, we determine the Gauss …
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.
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…
New systems of linear PDEs discovered in 3D contact manifolds.
problem Investigating linear PDEs of -type.
method Complete local classification using extrinsic geometry.
result 7 new systems of second-order linear PDEs with 8-dimensional solution spaces.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets .
result Generalized Geroch's monotonicity formula to initial data sets.
This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are…
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.
Improved neural PDEs trained on augmented data enhance model accuracy and efficiency.
problem Training neural PDEs on limited data to accurately represent complex systems.
method Space-filling sampling of local states to generate augmented training data.
result Data-augmented neural PDEs outperform traditional emulators in accuracy and stability.
EPGP priors solve linear PDEs from data.
problem Modeling physical systems with PDEs.
method EPGP priors based on Ehrenpreis-Palamodov principle.
result EPGP priors improve computation time and precision.
Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in .
method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
Finite-gap solutions approximate jets of initial data for certain BKM systems.
problem Approximating jets of initial data for specific PDE systems.
method Using finite-reduction map to finite-gap solutions of Stäckel systems.
result Full jet-surjectivity for KdV and Kaup--Boussinesq, partial for Camassa--Holm.
Graph neural networks learn PDEs from sparse, irregular data.
problem Learning PDEs from irregularly spaced data.
method Continuous-time differential model with graph neural networks for arbitrary discretizations.
result Efficient inference with continuous-time adjoint method.
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.
Finite-dimensional reductions and finite-gap type solutions of multicomponent integrable PDEsmath-ph
The paper solves integrable systems of PDEs, including famous equations.
problem Constructing solutions for multicomponent integrable PDEs.
method Reduction to a finite-dimensional system, using Nijenhuis geometry.
result Animations of multi-component soliton and cnoidal solutions.
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…
The paper studies third-order PDEs invariant under affine transformations and connects them to the Fubini-Pick invariant.
problem Investigating third-order PDEs invariant under affine transformations.
method Using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant PDEs on homogeneous manifolds].
result Derives third-order PDEs from the Fubini-Pick invariant.
New integrable systems derived from Nijenhuis geometry.
problem Developing new integrable systems from Nijenhuis geometry.
method Constructing a new series of multicomponent integrable PDE systems.
result Many famous integrable systems are contained within the new series.
Novel method controls complex physical systems over long time frames.
problem Controlling complex nonlinear physical systems over long time frames.
method Hierarchical predictor-corrector scheme with separate planning and control networks.
result Successfully controls complex physical systems like incompressible Navier-Stokes equations.
In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal th…
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.
A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquel…
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
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…
It is known that any maximal space-like surface without isotropic points in the four-dimensional pseudo-Euclidean space with neutral metric admits locally geometric parameters which are special case of isothermal parameters. With respect to such parameters the surface is determined uniquely up to a motion by the Gauss …
Automates discovering PDEs from data in dynamical systems.
problem Identifying PDEs from data in dynamical systems is challenging.
method ARGOS-RAL framework using sparse regression with recurrent adaptive lasso.
result ARGOS-RAL effectively identifies PDEs from noisy and non-uniformly distributed data.
One considers a special class of PDEs systems and one determines the associated symmetry group. Particulary, for the Blair system, one finds the symmetry group. A solutions of the Blair system gives a conformally flat contact metric structure and also it defines a "force-free" model of solar physics. By using the symme…
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
This is an elementary introduction to exterior differential systems motivated by two examples: minimal submanifolds and the isometric embedding problem. The two main goals of the lectures are: 1. To explain how to find an appropriate geometric setting for studying a given system of pde. 2. To explain the Cartan algorit…
Extracts intrinsic spatial coordinates for complex agent systems to learn PDEs.
problem Modeling collective dynamics of heterogeneous agents.
method Data-driven extraction of intrinsic spatial coordinates, learning PDEs in emergent space.
result Collective dynamics can be approximated through learned PDEs in emergent coordinates.
Two geometric realizations of Lie superalgebra found.
problem Geometric realization of Lie superalgebra.
method Explicit geometric realizations as supersymmetries of super-PDE systems.
result Two orders of super-PDE systems found for .
Explains Bernstein theorems for various geometric PDEs.
problem Bernstein problem for minimal surface, Monge-Ampère, and special Lagrangian equations.
method Expository review of existing theorems and systems.
result Discussion of Bernstein theorems for different geometric PDEs.
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental features in numerical PDEs and nonlinear analysis: irregular solutions. First, the S…
Study on stability of geodesic maps in non-isotropic manifolds.
problem Stability of totally geodesic wave maps in non-isotropic manifolds.
method Factorization property, PDE system in geodesic normal coordinates, global existence result via hyperboloidal foliation.
result Established global existence for small initial data, leading to geometric stability.
Invariant reduction preserves Poisson structures in PDEs.
problem Preserving Poisson structures in invariant solutions of PDEs.
method Invariant reduction applied to PDEs through Hamiltonian operators and Poisson bivectors.
result Inherited Poisson brackets match original systems up to sign.
New formula for portfolio risk management using conditional PDEs.
problem Optimal diversification and risk management of portfolios.
method Closed-form formula for conditional probability, Gaussian copulas, conditional risk-neutral PDE.
result Dynamic monitoring of portfolio volatilities and weights from PDEs.
The study describes metrics geodesically compatible with Nijenhuis operators and their applications to integrable systems.
problem Geodesically compatible metrics and their applications to integrable systems.
method Describes metrics geodesically compatible with a gl-regular Nijenhuis operator and shows how these metrics relate to integrable PDE systems.
result Every metric geodesically compatible with a Nijenhuis operator gives a finite-dimensional reduction of an integrable PDE system.
We study an interacting particle system in motivated by Stein variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic algorithm for sampling from a given probability density with unknown normalization. We prove that in the large particle limit the empirical measure of the particle s…
On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEsmath.DG
In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time--dependent symmetries. In particular we describe how that the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time…
Develops VAEs for learning complex physical systems from data.
problem Learning low-dimensional representations of nonlinear physical systems.
method Variational Autoencoders with manifold latent spaces.
result Effective in learning nonlinear Burgers equation and constrained mechanical systems.
Applications of differential algebra for computing Lie algebras of infinitesimal CR-automorphismsmath.RA
We perform detailed computations of Lie algebras of infinitesimal CR-automorphisms associated to three specific model real analytic CR-generic submanifolds in C^9 by employing differential algebra computer tools -- mostly within the Maple package DifferentialAlgebra -- in order to automate the handling of the arising h…