Deep Learning for Partial Differential Equations William Wei - - PowerPoint PPT Presentation

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May 09, 2023 39 likes •160 views

Deep Learning for Partial Differential Equations William Wei National Center for Supercomputing Applications Center for Artificial Intelligence Innovation Department of Physics, University of Illinois Urban-Champaign Reference:

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Deep Learning for Partial Differential Equations

William Wei

National Center for Supercomputing Applications Center for Artificial Intelligence Innovation Department of Physics, University of Illinois Urban-Champaign

Reference: http://www.dam.brown.edu/people/mraissi/research/1_physics_informed_neural_networks/

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ut + 𝒪[u; λ] = 0, x ∈ Ω, t ∈ [0,T]

is the solution to the equation.

u(t, x)

𝒪[u; λ] is a non-linear function of u(t, x) parameterized by λ

General formulation of partial differential equations (PDEs)

Data-driven solutions for PDEs
Data-driven discovery of PDEs

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Data-driven solutions for PDEs: Continuous Time Models

u(t, x) =

t x u

f := ut + 𝒪[u; λ]

ut + 𝒪[u; λ] = 0

Minimize the loss function:

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def u(t, x): u = neural_net(tf.concat([t,x],1), weights, biases) return u def f(t, x): u = u(t, x) u_t = tf.gradients(u, t)[0] u_x = tf.gradients(u, x)[0] u_xx = tf.gradients(u_x, x)[0] f = u_t + u*u_x - (0.01/tf.pi)*u_xx return f

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Example: Burgers

ut + uux − (0.01/π)uxx = 0, x ∈ [−1,1], t ∈ [0,1],

u(0,x) = − sin(πx) u(t, − 1) = u(t,1) = 0

u(t, x) ≈ ̂ uθ(t, x)

Neural network approximation:

f(θ) := ̂ ut + ̂ u ̂ ux − (0.01/π)uxx

L(θ) = ∥f(θ)∥2 + ∥ ̂ u(0,x) − u(0,x)∥2 + ∥ ̂ u(t, − 1)∥2 + ∥ ̂ u(t,1)∥2

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Data-driven solutions for PDEs: Discrete Time Models

Runge-Kutta methods:

un+ci = un − Δt

∑

j=1

aij𝒪[un+cj], i = 1,...,q

Euler method

ut + 𝒪[u; λ] = 0

un+1 = un − Δt𝒪[un; λ]

u(t) ≈ {u0, u1, . . . , un, . . . uN}

un+1 = un − Δt

∑

j=1

bj𝒪[un+cj]

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un

i := un+ci + Δt q

∑

j=1

aij𝒪[un+cj], i = 1,...,q

un

q+1 := un+1 + Δt q

∑

j=1

bj𝒪[un+cj]

un = un

i , i = 1,...,q

un = un

q+1

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multi-output neural network

un

i := un+ci + Δt q

∑

j=1

aij𝒪[un+cj], i = 1,...,q

un

q+1 := un+1 + Δt q

∑

j=1

bj𝒪[un+cj]

[un+c1(x), un+c2(x), . . . , un+cq(x), un+1(x)] [un

1(x), un 2(x), . . . , un q(x), un q+1(x)]

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un = un

i , i = 1,...,q

un = un

q+1

𝒫(Δt2q)

Temporal error accumulation :

Δt = 0.8, q = 500, Δt2q = 0.81000 ≈ 10−97

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