Solving high-dimensional partial differential equations using deep learning

Han, Jiequn; Jentzen, Arnulf; Weinan, E

doi:10.1073/pnas.1718942115

Cited by 1,470 publications

(1,155 citation statements)

References 28 publications

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“…Finally, developing an efficient numerical method for mean field games and mean field type control problems remains as a very important issue. For a general problem, due to its infinite dimensionality, machine learning techniques (such as in [16]) are promising candidates. If the problem can be approximated by a linear quadratic setup, its solution may help to accelerate the speed of convergence for the learning process in the spirit of the work [19].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

Fujii

2019

SSRN Journal

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In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well as the coefficient functions of the state dynamics are shared among the agents within each population, they can be different population by population. We study the mean field limit for the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) the agents in some populations are cooperative but those in the other populations are not. We provide several sets of sufficient conditions for the existence of a mean field equilibrium for each of these cases. Furthermore, under appropriate conditions, we show that the mean field solution to each of these problems actually provides an approximate Nash equilibrium for the corresponding game with a large but finite number of agents.Keywords : mean field game, mean field type control, FBSDE of McKean-Vlasov typewhere (X i,µ i ,µ j ) i∈{1,2},j =i are the solutions to the optimal control problems in (ii).Remark 3.1. It is just for convenience to use the common dimension d (as well as k for A i ) for both populations. Optimization for given flows of probability measuresThe main assumptions in this section are as follows: Assumption 3.1. (MFG-a) L, K ≥ 0 and λ > 0 are some constants. For 1 ≤ i ≤ 2, the measurable functionsand g i : R d × P 2 (R d ) 2 → R satisfy the following conditions: (A1) The functions b i and σ i are affine in (x, α) in the sense that, for any (t, x, µ, ν, α) ∈ [0, T ] ×

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Section: Conclusion and Discussionmentioning

confidence: 99%

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

Fujii

2019

SSRN Journal

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“…Machine learning for multiscale systems. Machine learning methods have recently permeated into composites research and materials design for example to enable the homogenization of representative volume elements with neural networks [94,63,65,55] or the solution of high-dimensional partial differential equations with deep learning methods [41,32,33,125,126]. Uncertainty quantification in material properties is also gaining relevance, with examples of Bayesian model selection to calibrate of strain energy functions [71,77] and uncertainty propagation with Gaussian processes of nonlinear mechanical systems [59,60,105].…”

Section: State Of the Artmentioning

confidence: 99%

Multiscale Modeling Meets Machine Learning: What Can We Learn?

Peng

Alber

Tepole³

et al. 2020

Arch Computat Methods Eng

245

103

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Machine learning is increasingly recognized as a promising technology in the biological, biomedical, and behavioral sciences. There can be no argument that this technique is incredibly successful in image recognition with immediate applications in diagnostics including electrophysiology, radiology, or pathology, where we have access to massive amounts of annotated data. However, machine learning often performs poorly in prognosis, especially when dealing with sparse data. This is a field where classical physics-based simulation seems to remain irreplaceable. In this review, we identify areas in the biomedical sciences where machine learning and multiscale modeling can mutually benefit from one another: Machine learning can integrate physics-based knowledge in the form of governing equations, boundary conditions, or constraints to manage ill-posted problems and robustly handle sparse and noisy data; multiscale modeling can integrate machine learning to create surrogate models, identify system dynamics and parameters, analyze sensitivities, and quantify uncertainty to bridge the scales and understand the emergence of function. With a view towards applications in the life sciences, we discuss the state of the art of combining machine learning and multiscale modeling, identify applications and opportunities, raise open questions, and address potential challenges and limitations. This review serves as introduction to a special issue on Uncertainty Quantification, Machine Learning, and Data-Driven Modeling of Biological Systems that will help identify current roadblocks and areas where computational mechanics, as a discipline, can play a significant role. We anticipate that it will stimulate discussion within the community of computational mechanics and reach out to other disciplines including mathematics, statistics, computer science, artificial intelligence, biomedicine, systems biology, and precision medicine to join forces towards creating robust and efficient models for biological systems.

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“…Theoretical pricing bounds have been proposed in the interesting paper [14] dealing with XVA problems. General methods have also been proposed by Pham et al [2,16,17], themselves inspired by the BSDE solver of Jentzen et al [13] who approximate the solution of some specific nonlinear partial differential equations in dimension up to 100. Although not written with the vocabulary of reinforcement learning, most of these papers share ideas with the methods discussed in the RL community.…”

Section: Introductionmentioning

confidence: 99%

Deep Reinforcement Learning for Market Making in Corporate Bonds: Beating the Curse of Dimensionality

Guéant

Manziuk

2019

Applied Mathematical Finance

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In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for a large number of bonds to asset managers from all around the globe. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. Useful models exist, most of them inspired by that of Avellaneda and Stoikov. These models describe the complex optimization problem faced by market makers: proposing bid and ask prices in an optimal way for making money out of the difference between bid and ask prices while mitigating the market risk associated with holding inventory. While most of the models only tackle one-asset market making, they can often be generalized to a multi-asset framework. However, the problem of solving numerically the equations characterizing the optimal bid and ask quotes is seldom tackled in the literature, especially in high dimension. In this paper, our goal is to propose a numerical method for approximating the optimal bid and ask quotes over a large universe of bonds in a model à la Avellaneda-Stoikov. Because we aim at considering a large universe of bonds, classical finite difference methods as those discussed in the literature cannot be used and we present therefore a discrete-time method inspired by reinforcement learning techniques. More precisely, the approach we propose is a model-based actor-critic-like algorithm involving deep neural networks.

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Solving high-dimensional partial differential equations using deep learning

Cited by 1,470 publications

References 28 publications

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

Multiscale Modeling Meets Machine Learning: What Can We Learn?

Deep Reinforcement Learning for Market Making in Corporate Bonds: Beating the Curse of Dimensionality

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