New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
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48 results for “Mixed PDE”
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.
Some recent developments on isometric immersions via compensated compactness and gauge transformsmath.AP
Survey on recent developments in isometric immersions using PDE techniques.
problem Analyzing isometric immersions with low Sobolev regularity.
method Compensated compactness and Coulomb-Uhlenbeck gauges.
result Weak continuity and stability of Gauss-Codazzi-Ricci equations.
Develops numerical methods for PDEs on hypergraphs and networks.
problem Solving PDEs on complex geometric structures like hypergraphs and networks.
method Hybrid finite element methods, focusing on hybrid discontinuous Galerkin methods.
result Derives numerical approximations for PDEs on hypergraphs and networks.
We develop a mixed least squares Monte Carlo-partial differential equation (LSMC-PDE) method for pricing Bermudan style options on assets whose volatility is stochastic. The algorithm is formulated for an arbitrary number of assets and volatility processes and we prove the algorithm converges almost surely for a class …
Novel IMEX scheme solves financial PDEs with mixed derivatives.
problem Numerical approximations for financial PDEs with mixed derivatives.
method Second order finite volume IMEX Runge-Kutta scheme.
result Achieves true second order convergence with non-regular initial conditions.
PDEs for pricing interest rate derivatives under the new generalized Forward Market Model (FMM)q-fin.PR
Derives PDEs for pricing RFR derivatives under a new FMM model.
problem Valuation of interest rate derivatives under a new FMM model.
method Develops PDEs and finite differences methods for numerical solution.
result First use of PDE methods for RFR derivatives valuation.
Path-dependent PDEs model VIX and Realised Variance options.
problem Modeling volatility derivatives with path-dependence.
method Continuous stochastic volatility model with Gaussian Volterra process, proving well-posedness of PDEs.
result Formulae for greeks and implied volatility provided, finite-dimensional pricing PDEs obtained in Markovian models.
The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.
problem Finding general solutions for Lie systems.
method Develops mixed superposition rules for Lie systems with imprimitive Lie algebras and semidirect sums.
result Extends coalgebra method to Lie systems of partial differential equations.
Researchers create solutions for naked singularities in Einstein vacuum equations.
problem Constructing solutions for the interior region of naked singularities in Einstein vacuum equations.
method Novel self-similarity and study of mixed degenerate elliptic-hyperbolic PDE's.
result Gluing together interior and exterior solutions produces a naked singularity.
In this paper, the valuation of European and path-dependent options in foreign exchange (FX) markets is considered when the currency exchange rate evolves according to the Heston model combined with the Cox-Ingersoll-Ross dynamics for the stochastic domestic and foreign short interest rates. The mixed Monte Carlo/PDE m…
Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We s…
Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
problem Index theory and analytic torsion of nonlinear PDEs.
method Microlocal sheaf theory, factorization algebras, Spencer hypercohomology.
result Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
Developed MF-PINNs to solve coupled Stokes-Darcy equations more accurately.
problem Solving coupled Stokes-Darcy equations with varying physical constants.
method Combining VP and SV forms with adjusted weights in MF-PINNs.
result Improved accuracy of streamline and pressure fields in numerical experiments.
A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from multimodal distributions. Due to metastability, multimodal distributions are difficult to sample using standard Markov chain Monte Carlo methods. We propose a new sampling algorithm based on a birth-death mechanis…
New algorithm solves complex equations using deep learning.
problem High-dimensional nonlinear PDEs and BSDEs.
method Iterated time discretization, deep neural networks, stochastic gradient descent.
result Increased accuracy and reduced complexity compared to existing methods.
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.
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.
We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points ar…
Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.
problem Existence of solutions to the Calabi-Yau equation for balanced metrics.
method Introduced a metric space of mixed-volume forms and derived a geodesic equation.
result Existence of solutions to the Calabi-Yau equation on all balanced manifolds.
Tradable Schemescond-mat.stat-mech
In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very e…
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…
We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough () coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restri…
PAGP uses physics-assisted Gaussian processes to solve and learn PDEs.
problem Solving and discovering unknown coefficients in PDEs with initial and boundary conditions.
method Physics-assisted Gaussian processes with continuous, discrete, and hybrid models.
result Effective in solving and discovering unknown coefficients in PDEs.
Paper proves short-time existence for network flow, providing detailed insights.
problem Short-time existence for the flow of a network of curves in the plane.
method Direct PDE approach, handling singularities at vertices using self-similar expanding solutions.
result Substantially more detailed information about network resolution into a regular one.
The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fieldsmath.FA
Computes indices of mixed order Dirac-type operators and related tensor fields.
problem Computing indices of mixed order Dirac-type operators and tensor fields.
method Using Hilbert complexes and differential operators of mixed order, computing indices with cohomology groups of tensor fields.
result Computation of indices for elasticity and biharmonic complexes.
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.
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.
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 …
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.
New methods solve SPDEs for financial derivative pricing.
problem Deriving the price of financial derivatives using SPDEs.
method Developed a conditional Feynman-Kac formula to solve SPDEs.
result Established new numerical methods for mixed Monte-Carlo PDEs.
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.
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.
The paper presents a PDE method for xVA incorporation in financial derivatives.
problem Incorporating value adjustments (xVA) in financial derivative pricing.
method Analytical solution of PDEs in the Black-Scholes framework.
result New semi-closed formulas for xVA are derived and compared to Monte-Carlo and numerical methods.
We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these net…
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.
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.
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under …
Meta-learning neural networks to solve diverse PDEs efficiently.
problem Efficiently solving new PDE problems with minimal training.
method Neural network meta-learning of PDE problem representations.
result Meta-learned neural networks predict PDE solutions with high accuracy.
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 …