We explore completely exceptional 2nd order scalar PDEs and their connection to Monge-Ampère equations.
problem Identifying a specific class of nonlinear PDEs that are not genuinely nonlinear.
method Unified geometric background review and definition of completely exceptional PDEs and Monge-Ampère equations.
result The class of completely exceptional 2nd order scalar PDEs reduces to Monge-Ampère equations.
In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the eq…
Study finds lowest-degree invariant PDEs over rational contact manifolds.
problem Finding the simplest invariant PDEs over rational contact manifolds.
method Analyzes simple Lie algebras to find lowest-degree invariant second-order PDEs.
result Provides explicit formulas and proves uniqueness for specific types of Lie algebras.
New dynamics derived from Lie groups using 2nd order tangent groups.
problem Deriving dynamics on complex Lie groups.
method Using double cross product groups and 2nd order Euler-Lagrange equations.
result 2nd order Lagrangian dynamics on double cross product groups derived.
The paper controls complex systems using energy-based methods.
problem Controlling nonlinear infinite-dimensional systems with dissipation.
method Energy-based control using Casimir functionals and energy balancing.
result Demonstrated control of a nonlinear Euler-Bernoulli beam.
Developed a BP algorithm for training neural networks with 2nd order neurons.
problem Training neural networks with nonlinear quadratic operations.
method Created a general backpropagation algorithm.
result Validated the generalized BP algorithm through numerical studies.
This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov.
Lecture notes on geometric structures of 2nd order ODEs.
problem Understanding the geometry of second-order ordinary differential equations.
method Construction of Cartan connection, computation of invariants, recognition of symmetric equations.
result Constructive aspects of local equivalence problem for 2nd order ODEs.
We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, L(X,∇u,D2u)=f, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nab…
The study examines how market trade randomness influences price and return volatility.
problem The accuracy of predicting market-based volatilities and macroeconomic variables is limited.
method Analyzes time series of trade values and volumes, and develops econometric methodologies for predicting volatilities.
result Current macroeconomic models underestimate the accuracy of predicting market-based volatilities and macroeconomic variables.
We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We also show that solutions exists globally on these sets. Finally, we establish the e…
Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of 2…
Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such c…
Study of geometry-induced potentials for curved regions in 3D space.
problem Finding a curved region with a prescribed geometry-induced potential.
method Formalism to deduce a meaningful Hamiltonian for confinement, solving for curves and surfaces using PDEs and ODEs.
result Existence of geometry-induced bound and localized states for helicoidal surfaces.
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.
Solved metrisability for Painlevé equations and their geodesics.
problem Metrisability of Painlevé equations and geodesics.
method Solved the metrisability problem for all 2nd order ODEs with Painlevé property.
result Integral curves of Painlevé equations are geodesics of a (pseudo) Riemannian metric.
The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…
Paper uses deep learning to solve PDEs without supervision.
problem Solving elliptic PDEs without labeled data.
method Uses deep neural networks and least-squares functionals.
result Demonstrates effectiveness on 1D second-order elliptic PDEs.
We prove new results on existence of solutions for the prescribed gaussian curvature problem on the euclidean sphere S^2. Those results are achieved by relating this problem with the holomorphic triples theory on Riemann surfaces. We think this approach might be applied to study some other semi-linear elliptic equation…
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 …
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of …
It is shown that the characteristic vector field associated to a first order PDE has the same form of an infinitesimal generator of an odd-symplectic transformation with contact Hamiltonian the given PDE. It is considered under which condition such PDE has a characteristic vector field commuting with a generator of an …
The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkma…
Develops methods for conditional symmetries of higher-order PDEs, removing unnecessary assumptions.
problem Formulating conditional symmetries for higher-order PDEs with unnecessary assumptions.
method Geometrical formulation and Lie systems approach to derive Lie algebras of conditional symmetries.
result New insights and methods for solving higher-order PDEs.
Paper formulates governing equations for membrane O surfaces.
problem Formulating equations for membrane O surfaces.
method Formulated governing equations for membrane O surfaces of the 1st and 2nd kind.
result Membrane O surfaces are a subclass of Demoulin's Ω surfaces.
We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…
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.
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
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.
Simplified proof for Frank and Lieb's inequality on Heisenberg group.
problem Proving the sharp Frank-Lieb inequality on the Heisenberg group.
method Simpler proof based on 2nd variation of subcritical functionals.
result A simpler proof of the inequality without the need for minimizer existence.
VarNet solves PDEs with deep neural networks using variational loss.
problem Solving partial differential equations (PDEs) efficiently and accurately.
method VarNet uses a novel variational loss function and optimizes space-time samples for training deep neural networks.
result VarNet models are smooth, differentiable, and directly usable for PDE control and optimization.
Study on 4D PDEs with half-flat conformal structure leading to Monge-Ampere equations.
problem Characterizing second-order PDEs in 4D with specific conformal structures.
method Analysis of Monge-Ampere property and dispersionless Lax pairs.
result All known scalar second-order integrable dispersionless PDEs in 4D are of Monge-Ampere type.
HTE improves PINNs for high-dimensional, high-order PDEs by reducing computational cost and memory usage.
problem Challenges in solving high-dimensional, high-order PDEs with PINNs due to computational cost and memory constraints.
method Introduces Hutchinson Trace Estimation (HTE) to transform Hessian matrix calculations into Hessian vector products (HVP), reducing computational cost and memory usage.
result HTE significantly reduces memory consumption and computational cost, enabling faster and more efficient solution of high-dimensional and high-order PDEs.
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.
Method finds explicit solutions to certain PDEs.
problem Finding solutions to specific types of PDEs.
method Exploiting solvable structures to find explicit solutions.
result Effectiveness demonstrated on several examples.
The paper proved that every C2-solution of a given first order PDEs system, regarded on the jet fibre bundle of order one J1(T,M), 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 …
New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
We analyze small price impacts in a multidimensional utility maximization problem using PDEs.
problem Small nonlinear price impacts in a multidimensional utility maximization problem.
method Asymptotic expansion using nonlinear PDEs related to ergodic control and linear parabolic PDEs.
result Leading order correction to the value function is characterized by a nonlinear second order PDE.
In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomol…
New method solves high-dimensional PDEs and 2BSDEs efficiently.
problem High-dimensional fully nonlinear PDEs and 2BSDEs in financial models.
method Connection between PDEs and 2BSDEs, merged formulation, temporal discretization, spatial approximation via neural nets, stochastic gradient descent.
result Efficient and accurate solution for high-dimensional nonlinear expectations.
The paper explores the geometry of Lagrangian Grassmannians and their connection to PDEs.
problem Understanding the geometric properties of Lagrangian subspaces.
method Thorough review of geometric properties and their relation to PDEs.
result Hypersurfaces in the Lagrangian Grassmannian correspond to second-order PDEs.
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this …
Geometric approach to PDEs using contact manifolds and Lagrangian Grassmannians.
problem Scalar PDEs in n variables of order one and two.
method Underlying (2n+1)-dimensional contact manifold and Lagrangian Grassmannian bundle.
result Introduction of geometric methods to PDEs, including scalar PDEs of order one and two.
S2GPT-PINNs solve PDEs with sparse, small models.
problem Efficiently solving parametric PDEs with minimal resources.
method Sparse and small architecture, mathematically rigorous greedy algorithm, knowledge distillation, down-sampling.
result Achieves high efficiency with significantly fewer parameters.
PDE-Net learns PDEs from data using neural networks.
problem Learning PDEs from complex system dynamics.
method Proposes PDE-Net, a feed-forward deep network to learn differential operators and nonlinear responses.
result PDE-Net can accurately predict dynamics and uncover hidden PDE models.
Generative network integrates into ROM for PDEs, matching measurements and estimating uncertainties.
problem Predicting and quantifying uncertainties in numerical simulations of PDEs.
method Generative network (GN) integrated into a reduced-order model (ROM) framework for inverse problems.
result GN-based ROM efficiently quantifies uncertainty and matches measurements with high accuracy.
Study shows rate of convergence for particle approximation of PDEs in Wasserstein space.
problem Analyzing convergence rates for particle approximations of PDEs in Wasserstein space.
method Backward stochastic differential equations techniques.
result Proved a rate of convergence of order 1/N for pathwise error and 1/sqrt(N) for L2-error on the derivative.
The hybrid Monte Carlo algorithm (HMCA) is applied for Bayesian parameter estimation of the realized stochastic volatility (RSV) model. Using the 2nd order minimum norm integrator (2MNI) for the molecular dynamics (MD) simulation in the HMCA, we find that the 2MNI is more efficient than the conventional leapfrog integr…