DeepSets and Their Derivative Networks for Solving Symmetric PDEs

Germain, Maximilien; Laurière, Mathieu; Pham, Huyên; Warin, Xavier

doi:10.1007/s10915-022-01796-w

Cited by 19 publications

(16 citation statements)

References 24 publications

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“…and similarly: Table 1. Comparison of the standard Monte Carlo estimator δ(N ) (17) with the importance sampling estimator δ(N ) (23) and their corresponding relative errors ρ(δ(N )) (51) and ρ( δ(N )) (52) with G as in Equation ( 47) and particles obeying (48) with initial condition y = 0.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In the mean-field setting, methods of stochastic control are already being used to address the issue of numerically constructing solutions to the HJB Equation (15), see e.g. [22,25,46,52,53,58,64]. Moreover, using the calculus of variations form of the rate function of Dawson-Gärtner [32], there are some examples in the literature where perturbation expansions have been made to approximate the optimal path of the controlled McKean-Vlasov Equation corresponding to certain types of rare events [12,47].…”

Section: Discussionmentioning

confidence: 99%

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Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Zachary¹,

Heldman²

2022

Preprint

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We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field (McKean-Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB Equation on Wasserstein Space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB Equation, our scheme can have vanishingly small relative error in the many particle limit.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Zachary¹,

Heldman²

2022

Preprint

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“…This yields an approach to solving the continuous space master equation described in Section 3.2.1.4 by first introducing a finite-state model that approximates the continuous model and then using the DGM method. Other methods could be investigated, such as the one proposed in [93]: a combination of dynamic programming, Monte Carlo simulations and symmetric neural networks is used to solve the Bellman equation arising in a continuous space MFC problem.…”

Section: The Function Hmentioning

confidence: 99%

Recent Developments in Machine Learning Methods for Stochastic Control and Games

Hu¹,

Laurière²

2023

Preprint

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In this paper, we give an overview of recently developed machine learning methods for stochastic control problems and games. The main focus is on deep learning methods that have unlocked the possibility to solve such problems even when the structure is complex or the dimension is high, which is not feasible with traditional numerical methods. The new approaches build on recent breakthrough machine learning methods for high-dimensional partial differential equations or backward stochastic differential equations, or on model-free reinforcement learning for Markov decision processes. This review summarizes state-of-the-art works at the crossroad of machine learning and stochastic control and games. It also discusses connections with real applications and identifies unsolved challenges.

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“…The introduction of the function b is actually motivated by numerical analysis. In fact it corresponds to the drift of training simulations for approximating the function , notably by machine learning methods, and should be chosen for suitable exploration of the state space; see the detailed discussion in our companion paper [15]. In this paper we fix an arbitrary function b (satisfying a Lipschitz condition to be made precise later).…”

Section: Particle Approximation Of Wasserstein Pdesmentioning

confidence: 99%

Rate of convergence for particle approximation of PDEs in Wasserstein space

2022

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We prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

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DeepSets and Their Derivative Networks for Solving Symmetric PDEs

Cited by 19 publications

References 24 publications

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Recent Developments in Machine Learning Methods for Stochastic Control and Games

Rate of convergence for particle approximation of PDEs in Wasserstein space

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