We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
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48 results for “complex partial differential equation”
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Survey explores cohomology's roles in applied math and sciences.
problem Understanding cohomology's role in solving differential equations.
method Examining differential complexes and structure-preserving discretizations.
result Various fundamental concepts in mechanics are formulated using differential complexes.
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.
The paper proves a Bernstein property for a specific complex partial differential equation.
problem Investigating the Bernstein property for a complex partial differential equation.
method Analyzing a fourth order complex partial differential equation with specific conditions.
result The equation has a Bernstein property under certain conditions.
Study deep neural nets for solving complex insurance equations.
problem Solving linear and semilinear parabolic PIDEs in high dimensions.
method Deep neural network algorithms for integro-differential equations.
result Viability of deep learning for solving high-dimensional PIDEs.
Neural networks can solve complex PDEs with minimal parameters.
problem Using neural networks to solve partial differential equations.
method Investigated two PDEs: Poisson and steady Navier--Stokes. Analyzed neural network architecture, initialization, loss function, and compared to classical methods.
result Small neural networks (<500 learnable parameters) can accurately solve complex PDEs.
Deep neural networks solve complex geometry PDEs.
problem Solving PDEs in complex geometries.
method Modified backpropagation for complex geometries, gradient descent or quasi-Newton optimization.
result Deep neural networks can approximate solutions to PDEs in complex geometries.
Survey of geometry developments, including complex structures on surfaces.
problem Enumerative geometry and complex structures on surfaces.
method Differential and algebraic geometry, nonlinear elliptic PDEs.
result Extensions to 4-manifolds and complex structures on surfaces of general type.
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as detour complexes. This elaborates on a lecture given at the IMA Summer Programme ``S…
New method learns physics equations from small data.
problem Learning partial differential equations from limited data.
method Hidden physics models using Gaussian processes.
result Data-efficient learning of complex equations.
FDNet learns PDEs from data with fast predictions.
problem Discovering complex systems behavior from data.
method Finite difference neural networks (FDNet) to learn PDEs from trajectory data.
result FDNet predicts future behavior with few trainable parameters.
New method solves complex curvature equations.
problem Solving semilinear scalar curvature equations.
method Mixed convex integration method.
result New proof of scalar curvature result.
Upper bounds on neural network complexity for PDE solutions.
problem Approximating solutions of parametric PDEs without knowing their exact form.
method Using low-dimensionality of solution manifolds and a small reduced basis.
result Neural networks can approximate PDE solutions with sizes dependent only on the reduced basis.
PIELM uses deep learning to solve PDEs quickly and accurately.
problem Solving partial differential equations (PDEs) efficiently and accurately.
method Physics Informed Extreme Learning Machine (PIELM) for solving PDEs.
result PIELM matches or exceeds the accuracy of Physics Informed Neural Networks (PINNs) on various problems.
Paper solves complex control problems using novel SDEs.
problem Solving stochastic differential games for nonlinear systems.
method Uses Deep Forward-Backward SDEs with neural networks.
result Numerical solution validated on two example systems.
Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketchingmath.NA
New method combines Monte Carlo and tensor networks for solving complex equations.
problem Solving high-dimensional partial differential equations efficiently.
method Uses Monte Carlo simulations and tensor train sketching for updates and re-estimations.
result Demonstrates versatility and efficacy in solving specific equations.
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
Solves division problem for L. Hörmander's systems.
problem Division problem for L. Hörmander's overdetermined systems.
method Formulates and proves divisibility criterion, coherence theorem.
result Establishes effective divisibility criterion and extends coherence theorem.
Some partial differential equations and conformal surfaces of the 4-dimensional Minkowski spacemath.DG
The paper develops complex representations for spacelike surfaces in 4D Minkowski space and solves associated PDEs.
problem Developing complex representations for spacelike surfaces in 4D Minkowski space.
method Introducing complex valued functions and using holomorphic and anti-holomorphic theory to solve PDEs.
result Explicit solutions for PDEs and characterization of conformal totally umbilical immersions.
Uniform estimates for complex equations on compact manifolds found.
problem Uniform estimates for form fully nonlinear PDEs on compact Hermitian manifolds.
method Local comparison with Monge-Ampère equations and finding an appropriate elliptic operator.
result A priori estimate for the equations.
Stochastic Partial Differential Equations on Evolving Surfaces and Evolving Riemannian Manifoldsmath.AP
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …
PDGM uses neural nets to solve complex financial equations.
problem Solving path-dependent partial differential equations (PPDEs)
method Generalized Deep Galerkin Method (PDGM) combining feed-forward and LSTM architectures
result PDGM successfully models solutions to various PPDEs, including financial derivatives.
Structure of a fourth-order dispersive flow equation through the generalized Hasimoto transformationmath.DG
Paper transforms a complex equation into simpler forms for analysis.
problem Analyzing a fourth-order dispersive flow equation on Kähler manifolds.
method Developed the generalized Hasimoto transformation to simplify the equation.
result Explicit expressions derived for three examples of compact Kähler manifolds.
New method solves PDEs for any initial condition without retraining.
problem Solving PDEs for different initial conditions requires retraining neural solvers.
method Formulate solution as conditional probability distribution.
result Approximates PDE solution for arbitrary initial conditions.
Deep reinforcement learning solves complex differential equations.
problem Solving nonlinear differential equations.
method Rule-based deep reinforcement learning approach.
result Solver captures intrinsic nature of equations with high accuracy.
Rust Implementation of Finite Element Exterior Calculus on Coordinate-Free Simplicial Complexesmath.NA
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
The paper introduces new functionals and equations for complex vector bundles.
problem Complex vector bundle equations and their solutions.
method Introducing generalized Donaldson's functionals and discussing their Euler-Lagrange equations.
result Uniqueness of solutions to complex vector bundle equations.
New quantum algorithm simplifies complex financial derivatives pricing.
problem Complex financial derivatives pricing with high dimensionality.
method Quantum-inspired variational algorithms combined with neural-network quantum states.
result Simplified pricing of European options with many correlated assets.
Study uses neural networks to solve complex equations efficiently.
problem Solving parametric partial differential equations.
method Machine learning and deep neural networks.
result Performance of the model is independent of parameter space dimension.
Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
Machine discovers PDEs from spatiotemporal data without prior knowledge.
problem Discovering PDEs from complex spatiotemporal data without prior knowledge.
method Sparse Spatiotemporal System Discovery () using Sparse Bayesian Learning.
result Automatically discovers ten types of PDEs from simulation data.
Clarifies when certain stochastic PDEs have affine solutions.
problem Existence of affine realizations for semilinear SPDEs driven by Lévy processes.
method Analyzes conditions for affine solutions to SPDEs driven by Lévy processes.
result Conditions for the existence of affine realizations are established.
New proof shows all conformal vector fields on complex hyperbolic space are Killing.
problem Proving all conformal vector fields on complex hyperbolic space are Killing.
method Local, analytic, and constructive approach using Lie group model and partial differential equations.
result Every conformal vector field on complex hyperbolic space is Killing.
Deep learning solves complex volatility equations.
problem Solving path-dependent PDEs in rough volatility.
method Interpreting PDE as BSDE, using neural network reservoir approach.
result Proved theoretical convergence for least-square regression.
Deep neural networks solve high-dimensional PDEs without explicit grids.
problem Solving high-dimensional PDEs using classical methods is computationally infeasible.
method Approximate solution with a deep neural network trained via FBSDEs.
result Deep learning can solve high-dimensional PDEs efficiently.
Geometric methods solve differential equations by analyzing space dimensions.
problem Interplay between geometry and partial differential equations.
method Calculating space dimensions associated with differential equations' zeros.
result Classical algebraic geometry results are central to analysis.
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …
Deep-MacroFin uses neural networks to solve complex economic models efficiently.
problem Solving high-dimensional partial differential equations in continuous time economics.
method Leverages deep learning, specifically Multi-Layer Perceptrons and Kolmogorov-Arnold Networks, optimized with HJB equations.
result Offers a more efficient solution (5 less memory, 40 fewer FLOPs) for 50D economic models.
Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Samplingstat.ML
Efficiently samples complex distributions using tensor train format.
problem Sampling from high-dimensional complex probability densities efficiently.
method Integrates tensor train format with backward stochastic differential equations (BSDEs) for fast, robust, and accurate sampling.
result Improved efficiency in sampling from challenging target distributions.
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.
The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…
We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
Clarifies when solutions to stochastic PDEs stay near given subsets.
problem Understanding the proximity of solutions to stochastic PDEs to given subsets.
method Analyzes distance between closed sets and solutions to stochastic PDEs.
result Clarifies conditions for solutions to stay near given subsets.
Notes on relative algebroidsmath.DG
Notes on relative algebroids for geometric problems.
problem Geometric problems and their solutions.
method Explains how relative algebroids arise from geometric problems and introduces their structural theory.
result Relative algebroids unify Lie algebroids with partial differential equations.
Affine realizations with affine state processes for stochastic partial differential equationsmath.PR
Clarifies when certain stochastic PDEs have affine state processes.
problem Characterizing stochastic PDEs with affine state processes.
method Characterization of initial points for affine realizations.
result Characterizes the set of initial points for affine realizations.
New method to write Dirac equation in curved spacetime.
problem Writing Dirac equation in curved spacetime using geometric concepts.
method Using analysis of partial differential equations instead of geometry.
result A non-geometric representation of the Dirac equation in curved spacetime.