Neural Galerkin schemes use active learning to solve high-dimensional equations.
problem Inaccurate function approximations in high dimensions with limited training data.
method Neural Galerkin schemes based on deep learning with active learning for high-dimensional PDEs.
result Active data collection improves the numerical solution of high-dimensional equations.
Paper explores solving HJB equations using neural networks.
problem Solving high-dimensional time-dependent HJB equations.
method Neural Galerkin methods with nonlinearly parametrized trial functions.
result Closed-form solutions for trial functions.
Galerkin method outperforms graph-based methods in spectral decompositions.
problem Improving spectral decomposition methods in machine learning.
method Restricting study to a small set of test functions using the Galerkin method.
result Statistical and computational superiority of Galerkin method over graph-based approaches.
A new method uses neural networks to improve POD-Galerkin models for complex systems.
problem Improving computational efficiency and accuracy in solving non-linear high-dimensional systems.
method Deep learning-based closure modeling using neural networks to approximate POD-Galerkin operators.
result The CD-ROM approach produces more accurate and stable models for complex systems.
Deep Galerkin Method estimates value function for mean-field control problem.
problem Optimal control of agents with average welfare as the objective.
method Apply DGM to estimate value function and distribution evolution.
result Neural network approximations converge to analytical solution.
New method learns diffusion transition density for Bayesian inference.
problem Bayesian inference on diffusions with inaccessible boundaries.
method Neural Galerkin framework to solve FP equation with Dirac mass.
result Approximates likelihood function for efficient posterior sampling.
New ARIMA framework improves forecast accuracy for economic and financial time series.
problem Improving forecast accuracy for nonlinear dynamics in time series data.
method Projection-based ARIMA framework using Galerkin basis expansions.
result Galerkin-SARIMA matches or improves forecast accuracy compared to classical ARIMA/SARIMA.
Review and compare model order reduction methods for process engineering.
problem Creating computationally efficient yet accurate models for real-time applications.
method Nonlinear model order reduction methods, including general-purpose and tailored approaches for chemical processes.
result Comparison of eight model order reduction methods applied to an air separation process model.
DGNet solves complex dynamical systems with neural networks and constraints.
problem Real-time accurate solutions for large-scale complex systems.
method Model-constrained discontinuous Galerkin Network (DGNet) for compressible Euler equations.
result DGNet achieves out-of-distribution generalization and improved stability.
New method uses randomized sparse neural networks to solve time-dependent PDEs more accurately and efficiently.
problem Numerical challenges in training neural networks sequentially in time to solve time-dependent PDEs.
method Introduces Neural Galerkin schemes that update randomized sparse subsets of network parameters at each time step.
result Up to two orders of magnitude more accurate and two orders of magnitude faster than dense update schemes.
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since mesh…
One popular approach to option pricing in Lévy models is through solving the related partial integro differential equation (PIDE). For the numerical solution of such equations powerful Galerkin methods have been put forward e.g. by Hilber et al. (2013). As in practice large classes of models are maintained simultaneous…
Study methods to recover unknown processes in PDEs from data.
problem Identifying unknown processes in time-dependent PDEs using observational data.
method Theoretical analysis and numerical approaches including Galerkin and collocation algorithms.
result The Galerkin algorithm is more suitable for practical situations with noisy data.
New methods for clustering graphs using spectral analysis.
problem Graph clustering for complex systems.
method Transfer operators and spectral properties.
result Spectral clustering can be interpreted using Koopman operators.
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to uni…
Paper establishes convergence rates for learning elliptic pseudo-differential operators.
problem Learning elliptic pseudo-differential operators in partial differential equations.
method Wavelet-Galerkin framework, structured infinite-dimensional regression problem, sparse estimator, matrix compression, nested-support strategy.
result Obtained convergence rates for the estimator and efficient Galerkin solver.
Deep learning method proves convergence for high-dimensional PDEs.
problem Solving high-dimensional nonlinear PDEs for mean field control problems.
method Deep Galerkin method (DGM) for Hamilton-Jacobi-Bellman (HJB) equations.
result DGM converges to the true value function of mean field control problems.
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.
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.
Research improves pricing of multidimensional American options using neural networks.
problem Pricing multidimensional American options efficiently and accurately.
method Time Deep Gradient Flow (TDGF) method and Deep Galerkin Method (DGM).
result TDGF method achieves high accuracy and faster training than DGM.
This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundar…
Proves regularity for quasilinear elliptic equations in metric spaces.
problem Regularity of quasilinear elliptic equations in metric measure spaces.
method Galerkin's method as an alternative to difference quotients.
result Second-order and Lipschitz regularity for a wide class of elliptic equations.
CoLoRA models predict PDE solutions quickly and accurately with minimal data.
problem Efficiently modeling PDE solutions with limited data.
method Continuous low-rank adaptation of neural networks trained on offline data.
result Predictions are orders of magnitude faster and more accurate than classical methods.
This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value R0=r0∈R, where θ∈R and σ>0 are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\…
Two neural network methods solve the master equation for MFGs.
problem Approximating Nash equilibria in stochastic, finite-agent games.
method Backward induction and direct PDE tackling neural networks.
result Neural networks can approximate the master equation's solution.
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a Petrov-Galerkin version of PINNs based on the nonlinear approximation of deep neur…
A machine learning approach to compute Black-Scholes prices with uncertain volatility.
problem Approximating financial markets with continuous-time models like Black-Scholes when data is discrete.
method Generalized Polynomial Chaos (gPC) method combined with a machine learning technique called Bi-Fidelity.
result Efficient numerical method to quantify uncertainty in derivative pricing.
Rapid simulations of advection-dominated problems are vital for multiple engineering and geophysical applications. In this paper, we present a long short-term memory neural network to approximate the nonlinear component of the reduced-order model (ROM) of an advection-dominated partial differential equation. This is mo…
A new model captures forward curve dynamics with stochastic volatility.
problem Modeling continuous-time evolution of forward curves in financial markets.
method Affine stochastic volatility model with modulated dynamics.
result Model allows for maturity-specific risk and volatility clustering.
Study optimal semi-static hedging for illiquid markets using dynamic cash and static quoted derivatives.
problem Optimal pricing of exotic derivatives in illiquid markets with bid-ask spreads.
method Use Galerkin method and integration quadratures to approximate hedging problem as convex optimization, solved by interior point method.
result Semi-static hedging improves pricing and reduces transaction costs compared to static or dynamic trading alone.
Book introduces deep learning methods with math, theory, and applications.
problem Understanding deep learning algorithms and their mathematical foundations.
method Reviews various ANN architectures and optimization methods, covers theoretical aspects.
result Provides a solid mathematical foundation for deep learning.
In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre p…
We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in Rn. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives…
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…
This research analyzes deep PDE solvers for option pricing accuracy.
problem Understanding the accuracy of deep learning methods for solving PDEs in option pricing.
method Comparative experiments with two neural network algorithms in Black--Scholes and Heston models.
result Empirical convergence rates and training times of TDGF method determined.
We derive an equation of motion for interest-rate yield curves by applying a minimum Fisher information variational approach to the implied probability density. By construction, solutions to the equation of motion recover observed bond prices. More significantly, the form of the resulting equation explains the success …
New deep learning method for option pricing in jump-diffusion models.
problem Option pricing in jump-diffusion models with high-dimensional assets.
method Implicit-explicit minimizing movement time-stepping approach using deep ANNs.
result Consistent and asymptotically correct solutions for large underlyings.
The objective of this paper is to investigate how noisy and incomplete observations can be integrated in the process of building a reduced-order model. This problematic arises in many scientific domains where there exists a need for accurate low-order descriptions of highly-complex phenomena, which can not be directly …
This paper simplifies hedge ratios in financial models using pathwise algorithmic differentiation.
problem Expensive and unstable computation of hedge ratios from pathwise sensitivities.
method Develops reduced stochastic hedge ratios of the form φ_j^r = Σ_j^r ξ_j^q X_q, retaining sensitivity tensor through empirical averages.
result Two coefficient criteria are introduced to minimize pathwise residuals and satisfy moment equations.
The most recent update of financial option models is American options under stochastic volatility models with jumps in returns (SVJ) and stochastic volatility models with jumps in returns and volatility (SVCJ). To evaluate these options, mesh-based methods are applied in a number of papers but it is well-known that the…
Nonlocal Bayesian modeling for continuous spatio-temporal dynamics
problem Handling irregular time points, sparse observations, and nonlocal interactions in spatio-temporal forecasting
method Hierarchical Bayesian framework with coordinate-based spatial basis expansion and continuous-time ODE
result Strong forecasting and uncertainty calibration
This paper presents four different ways of looking at the well-known Least Squares Temporal Differences (LSTD) algorithm for computing the value function of a Markov Reward Process, each of them leading to different insights: the operator-theory approach via the Galerkin method, the statistical approach via instrumenta…
Paper solves investment strategy optimization with deep learning.
problem Maximizing investor utility with optimal asset allocation.
method Solves PDEs with Deep Galerkin method.
result Deep learning algorithm outperforms finite difference method.
Optimizes trading in CFMMs and exchanges using deep learning.
problem Optimizing trading strategies in CFMMs and exchanges.
method Develops a model accounting for interaction between CFMMs and exchanges, employs deep Galerkin method to solve dynamic programming equation.
result Optimal strategy outperforms naïve strategies and is not prone to price slippage.
New rates for GLD and SGLD in infinite-dimensional spaces without dimensionality issues.
problem Gradient Langevin dynamics and SGLD convergence rates in high-dimensional spaces.
method Analysis of GLD and SGLD in infinite-dimensional Hilbert spaces, using stochastic differential equations and Markov chains.
result Derivation of dimension-free convergence rates for GLD and SGLD.
Derivative-informed models improve financial surrogates for accurate hedging and risk management.
problem Developing fast surrogate models for financial derivatives and risk quantities.
method Derivative-informed operator learning framework combining neural operators, random features, and tangent sensitivity equations.
result The framework reduces hedging and risk errors by 40-76% compared to standard surrogates.
Paper uses autoencoders for efficient reduced-order modeling of eigenvalue problems.
problem Efficiently modeling eigenvalue problems in high dimensions.
method Autoencoder-based reduced-order modeling for eigenvalue problems.
result Autoencoder-based models outperform standard POD-Galerkin methods in neutron diffusion applications.
We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportun…