A mesh-free method solves continuum-marginal optimal transport problems.
problem Recovering minimum-energy velocity fields from time-continuous probability marginals.
method Embeds weak continuity equation in a reproducing kernel Hilbert space, optimizing with mini-batch stochastic methods.
result Accurately recovers drift and maintains marginal consistency in synthetic experiments.
New method uses dynamic sampling to improve PINNs efficiency.
problem Improving sample efficiency and performance of PINNs.
method pdPINN, inspired by Eulerian formulation, uses dynamic Monte Carlo sampling from particle positions.
result Higher sample efficiency and improved performance of PINNs.
We review a numerical technique, referred to as the Transport-based Mesh-free Method (TMM), and we discuss its applications to mathematical finance. We recently introduced this method from a numerical standpoint and investigated the accuracy of integration formulas based on the Monte-Carlo methodology: quantitative err…
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.
Our paper aims to model supply and demand curves of electricity day-ahead auction in a parsimonious way. Our main task is to build an appropriate algorithm to present the information about electricity prices and demands with far less parameters than the original one. We represent each curve using mesh-free interpolatio…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. Th…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for these problems based on a deep learning approach. Specifically, the random PDE is…
A neural network method tackles high-dimensional diffeomorphic mapping problems.
problem High-dimensional diffeomorphic mapping struggles with the curse of dimensionality.
method Combines variational principles with quasi-conformal theory for accurate, bijective mappings.
result Validated accuracy, robustness, and effectiveness in complex registration scenarios.
SPH-ParVI uses fluid dynamics to sample unknown densities efficiently.
problem Sampling partially known densities or using gradients in probabilistic models.
method Smoothed Particle Hydrodynamics (SPH) for modeling fluid dynamics to approximate target densities.
result SPH-ParVI provides fast, flexible, scalable, and deterministic sampling for Bayesian inference and generative models.
We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demons…
New neural network solves Nirenberg problem for curvature on sphere.
problem Prescribing Gaussian curvature on S2 for metrics conformal to the round metric. method Mesh-free physics-informed neural network (PINN) that directly parametrises the conformal factor.
result Neural network achieves very low losses for realisable curvatures, distinguishing them from non-realisable ones.
A neural atlas simplifies 3D geometry simulation by avoiding meshing.
problem Simulation of complex 3D geometries with thin features or non-trivial topology.
method Learned geometric representation of overlapping volumetric coordinate charts, trained from point-cloud or level-set data.
result The learned atlas enables different solvers without re-meshing or re-parametrization.
FlowKac solves high-dimensional Fokker-Planck equations efficiently.
problem Intractability of Fokker-Planck equation solutions in high dimensions.
method Reformulates Fokker-Planck using Feynman-Kac, adaptive stochastic sampling, and normalizing flows.
result Significant computational efficiency and accuracy improvements over existing methods.
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal execution, mean field games, etc. The main idea behind DGM is to represent the unknown…
New method uses Riemannian geometry to describe molecular shapes.
problem Predicting drug-like molecules using shape similarity.
method Riemannian geometry applied to molecular surfaces.
result RGMolSA method captures molecular shape effectively.
This thesis advances algorithms and software for QMC, GP, and sciML.
problem Efficient high-dimensional integration, interpolation, and PDE modeling.
method Developed new algorithms and software for QMC, GP, and sciML.
result Efficient and accurate methods for high-dimensional problems.
Scalable solver reduces PDE uncertainty with active learning.
problem High computational cost in solving PDEs.
method Stochastic dual descent and clustering-based active learning.
result Solver scales to large number of collocation points.
MUSIC learns coupled systems with sparse data and incomplete physics.
problem Learning coupled systems with incomplete physical constraints and missing data.
method Sparsity induced multitask neural network framework integrating partial physical constraints with data-driven learning.
result MUSIC accurately learns solutions to complex coupled systems under data-scarce and noisy conditions.
Paper introduces FDM for efficient training of Neural SDEs.
problem Training Neural SDEs using existing methods is computationally expensive and unstable.
method Developed a novel scoring rule called Finite Dimensional Matching (FDM) to bypass signature kernels and reduce training complexity.
result FDM achieves superior performance in terms of computational efficiency and generative quality.
New deep learning method approximates Benes filter model.
problem Approximating high-dimensional SPDEs for filtering.
method Deep learning mesh-free neural network representation.
result First study of neural network method for Benes model.
This work develops machine learning for micromagnetic energy minimization.
problem Minimizing Gibbs free energy in full 3D micromagnetic simulations.
method Advanced machine learning techniques, including Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs), with reformulated bounds and optimization schemes.
result Competitive performance of machine learning methods compared to traditional numerical approaches.
Deep learning solves dynamic programming with recursive utility.
problem Challenges in solving high-dimensional discrete-time dynamic programming problems with recursive utility.
method Certainty Equivalent Learning (CEL) algorithm that learns certainty-equivalent value directly with neural networks.
result Accurate value and policy approximations in high-dimensional problems, comparable to VFI in some cases.
Paper introduces non-adversarial training for Neural SDEs using signature kernel scores.
problem Stability and mode collapse issues in adversarial training of Neural SDEs.
method Uses signature kernel scores as objective function for non-adversarial training.
result Non-adversarial training leads to better performance and more stable models.
Paper develops a new method for solving complex problems in generative modeling and mean-field games.
problem Solving complex problems in generative modeling and mean-field games.
method Reinterpreting Generalized Schrödinger Bridges (GSBs) as probabilistic models and using the nonlinear Feynman-Kac lemma.
result Demonstrates the efficacy of the new method in generative modeling and mean-field games.
Unified framework for Bayesian PDE-constrained inversion using physics-informed neural networks.
problem Incorporating prior distributions in function space into Bayesian PINN-based inversion.
method Functional-prior-based approaches (fpBPINN) to Bayesian PDE-constrained inversion using physics-informed neural networks (PINNs). Two complementary approaches: FPI-BPINN and fParVI-PINN.
result Accurate estimation of posterior distributions in seismic traveltime tomography and Darcy-flow permeability inversion.
A neural network solves Black-Scholes PDE for option pricing with uncertainty quantification.
problem Solving the Black-Scholes equation for option pricing with uncertainty.
method Physics-informed neural network (PINN) that embeds BS operator and conditions, handles early exercise via relaxation, and uses anchored-ensemble fine-tuning for uncertainty quantification.
result The method achieves low errors and accurate predictions for European and American options, outperforming data-driven baselines.
New method solves high-dimensional PDEs fast using physics-informed neural networks.
problem High computational cost in solving high-dimensional PDEs.
method Stochastic Dimension Gradient Descent (SDGD) for physics-informed neural networks (PINNs).
result Solves many high-dimensional PDEs including HJB and Schrödinger equations in 100,000 dimensions in 12 hours.
Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.
problem Constructing kernels for transport equations and Koopman eigenfunctions.
method Three methods: variational principle, Green's function, and resolvent operator.
result Kernels constructed via these methods are identical under mild assumptions.
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degr…
Develops weak PINNs for efficient manifold solutions of hyperbolic equations.
problem Challenges in approximating weak solutions of nonlinear hyperbolic equations on manifolds.
method Introduces a novel weak PINN (wPINN) formulation on manifolds leveraging well-posedness theory.
result Demonstrates efficient approximation of entropy solutions on manifolds with a complexity independent of ambient space dimension.