Solves natural PDE system for minimal surfaces in 4D Euclidean space.
problem Determining minimal surfaces in 4D Euclidean space.
method Explicitly solves the system of natural PDE's using two holomorphic functions.
result Expresses any solution of the system of natural PDE's by two holomorphic functions in the Gauss plane.
We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fra…
Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in R24. method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
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 …
Solves PDE system for minimal space-like surfaces in Minkowski space-time.
problem Solving the system of natural PDE's for minimal space-like surfaces.
method Using canonical Weierstrass representations, solves the system explicitly.
result Expresses solutions by means of two holomorphic functions.
The paper provides a Weierstrass representation for maximal space-like surfaces in 4D pseudo-Euclidean space.
problem Characterizing maximal space-like surfaces in 4D pseudo-Euclidean space.
method Developed a Weierstrass representation for maximal space-like surfaces using special parameters and holomorphic functions.
result Explicit solutions to the system of natural PDE's are found using holomorphic functions.
Enhances neural network solvers for PDEs with complex boundary conditions.
problem Challenges in solving PDEs with high accuracy and complex boundary conditions.
method Integrates natural gradient optimization with numerical time-stepping schemes to enforce Dirichlet boundary conditions.
result Superior accuracy and computational efficiency of the proposed methods for solving PDEs.
Automated PDE discovery from multiple noisy experiments.
problem Inherent variability in experiments makes single experiment inference unreliable.
method Randomised adaptive group Lasso sparsity estimator in deep learning framework.
result More generalizable PDEs found from multiple datasets.
Study of nonlinear PDEs using derived geometry and BV formalism.
problem Understanding non-linear PDEs via derived geometric methods.
method Derived enhancement of de Rham complex, algebro-geometric techniques, BV formalism.
result Natural derived enhancement of de Rham complex for nonlinear PDEs.
The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.
problem Symmetry phenomena in solutions of semilinear PDEs on Riemannian domains.
method General framework for formulating the symmetry problem; evidence from stable solutions; consideration of manifolds with density.
result Evidence that the framework is natural, with results for stable solutions.
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimens…
We smooth out the math of fitting surfaces into 3D space.
problem Fitting surfaces into 3D space isometrically.
method Introduce elliptic regularization to the PDE system.
result The regularization leads to a natural variational interpretation.
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.
New method solves high-dimensional Kolmogorov PDEs without curse of dimensionality.
problem Solving high-dimensional Kolmogorov PDEs efficiently and accurately.
method Deep learning-based numerical approximation method.
result Effective numerical approximation of Kolmogorov PDEs in high dimensions.
We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent …
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.
Improved flatness in annuli using PDE methods.
problem Flatness improvement in annuli.
method PDE-based approach adapted to exterior domains.
result Alternative proof of minimal surface end-structure and asymptotics.
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.
We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…
Physics-informed WNO learns PDE solutions without labeled data.
problem Data-hungry nature of WNO framework.
method Physics-informed WNO for learning PDE solutions.
result Validated and illustrated with four nonlinear systems.
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.
Paper studies shock wave transitions and completely exceptional PDEs via conformal geometry.
problem Understanding shock wave transitions and completely exceptional PDEs.
method Recasting conditions in terms of characteristics, embedding in Lagrangian Grassmannian, using BGG resolution.
result Completely exceptional PDEs, including Monge-Ampere, can be described via conformal geometry and BGG resolution.
Develops a new approach to describe gauge theories with background fields using presymplectic structures.
problem Describing gauge theories with background fields using presymplectic structures.
method Extension of the presymplectic BV-AKSZ approach to include background fields.
result Gauge theories with background fields correspond to presymplectic gauge PDEs over gauge PDEs describing background fields.
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. 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 …
The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.
problem Characterizing when numerical criteria for PDE solvability fail.
method Finite number of subvarieties violating Nakai type criterion, and their rigidity.
result Finite number of subvarieties violating the Nakai type criterion, and these subvarieties are rigid.
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.
Paper extends neural network method to irregular solutions in PDEs.
problem Solving irregular and data-enriched PDEs.
method Deep neural networks for numerical PDE solutions, extending to irregular and data-enhanced cases.
result Demonstrates ease and integration of large datasets in PDE modeling.
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.
This study uses machine learning to solve PDEs in engineering problems.
problem Solving Partial Differential Equations (PDEs) in engineering for precise system behavior.
method Deep Neural Networks (DNNs) for function approximation of PDE solutions.
result DNNs can effectively approximate PDE solutions for mechanical problems.
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
problem Formalizing and simplifying the study of partial differential equations (PDEs).
method Using cohomology of diffieties to formalize PDEs and their properties.
result Differential calculus on PDE solution spaces is homotopy calculus on horizontal De Rham algebras of diffieties.
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.
The paper finds a Weierstrass representation for a specific type of Lorentzian minimal surface.
problem Minimal Lorentzian surfaces in R24 with certain curvature conditions. method Weierstrass representation with respect to isothermal and canonical parameters.
result Explicit solution to the system of natural PDEs for general type surfaces.
Distance between evolving hypersurfaces is a PDE solution.
problem Tracking the distance between evolving hypersurfaces.
method Elliptic and parabolic PDEs, mean curvature flow.
result Local Harnack inequalities for the distance between evolving hypersurfaces.
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.
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In parti…
Bayesian PINNs optimize loss weights for PDEs and data.
problem Optimizing loss weights in physics-informed neural networks.
method Laplace approximation for efficient model evidence computation.
result Unified Bayesian setting for PDEs and noisy measurements.
A method constructs invariant PDEs on homogeneous manifolds.
problem Finding invariant PDEs on homogeneous manifolds.
method Describes a general method for constructing invariant PDEs by reducing the problem to invariant hypersurfaces under the action of the stability subgroup.
result Describes invariant PDEs for hypersurfaces in Euclidean and conformal spaces.
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
Probabilistic method combines space and time uncertainties in PDEs.
problem Separate treatment of space and time in PDE solvers obscures interactions and error quantification.
method Gaussian process interpretation of finite difference methods interacting with probabilistic ODE solvers.
result Joint quantification of space- and time-uncertainty possible without sacrificing ODE solver performance.
Efficient PDE method calibrates local volatility with stochastic interest rates.
problem Calibrating local volatility models with stochastic interest rates is time-consuming.
method Developed a PDE approach using ADI method to solve the forward equation.
result Effective and sufficient information for calibration and pricing is provided.
Derives PDEs from data using manifold learning and neural networks.
problem Identifying PDEs from unknown variables and dynamics.
method Combines manifold learning (Diffusion Maps) and neural networks.
result Emergent space identification connects with multiscale computation.
New method prices interest rate derivatives without Monte Carlo, achieving high accuracy and speed.
problem Arbitrage-free pricing of path-dependent interest rate derivatives using infinite-dimensional models.
method Casting the stochastic pricing problem as a deterministic PDE solved by FINNs, which minimize violations of the PDE and boundary conditions.
result FINNs achieve pricing accuracy within 0.04 to 0.07 cents per dollar of contract value compared to Monte Carlo benchmarks.
Proves existence of hypersurfaces with specific curvature properties.
problem Existence of hypersurfaces with prescribed boundary and Weingarten curvature.
method Solves a fully nonlinear elliptic PDE with a priori estimates.
result Proves existence of hypersurfaces under natural assumptions.
For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.
This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a…
Explores the geometric aspects of the Calabi Conjecture using PDE theory.
problem Existence of specific metrics in geometry.
method Nonlinear Elliptic PDE theory, focusing on Yau's solution.
result Proves the existence of a class of metrics with deep physical implications.
BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.
problem Enforcing boundary conditions in neural networks for PDE solutions.
method Boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (BEKAN) with radial basis functions (RBFs). Incorporates Dirichlet, periodic, and Neumann conditions.
result BEKAN outperforms MLP and B-splines KAN in solving PDEs with boundary conditions.