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Stochastic Differential Equations, Deep Learning, and High-Dimensional PDEs

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Now we will suss out the relationship between SDEs and PDEs and how this is used in scientific machine learning to solve previously unsolvable problems with a neural network as the intermediate. The easiest way to understand an SDE is by looking at it computationally. For the SDE defined as: d X t =f( X t ,t)dt+g( X t ,t)d W t 𝑑 𝑋 𝑡 = 𝑓 ( 𝑋 𝑡 , 𝑡 ) 𝑑 𝑡 + 𝑔 ( 𝑋 𝑡 , 𝑡 ) 𝑑 𝑊 𝑡 the "Euler method for SDEs", also known as Euler-Maruyama, is given by: X n+1 = X n +f( X n , t n )h+sqrt(h)g( X n , t n )ζ 𝑋 𝑛 + 1 = 𝑋 𝑛 + 𝑓 ( 𝑋 𝑛 , 𝑡 𝑛 ) ℎ + 𝑠 𝑞 𝑟 𝑡 ( ℎ ) 𝑔 ( 𝑋 𝑛 , 𝑡 𝑛 ) 𝜁 where ζ N(0,1) 𝜁   𝑁 ( 0 , 1 ) is a standard normal random variable (randn()). Thus a stochastic differential equation is an ordinary differential equation with a small stochastic perturbation at every point in time (continuously!). There are many definitions of Brownian motion. One of the easiest to picture is that it's the random walk where at every infinitesimal dt 𝑑 𝑡 you move by

Stochastic Differential Equations, Deep Learning, and High-Dimensional PDEs Stochastic Differential Equations, Deep Learning, and High-Dimensional PDEs Chris Rackauckas January 18th, 2020 Now we will suss out the relationship between SDEs and PDEs and how this is used in scientific machine learning to solve previously unsolvable problems with a neural network as the intermediate. What is an SDE? The easiest way to understand an SDE is by looking at it computationally. For the SDE defined as: \[ dX_t = f(X_t,t)dt + g(X_t,t)dW_t \] the "Euler method for SDEs", also known as Euler-Maruyama, is gi

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