On Hamilton-Jacobi partial differential equation and architectures - - PowerPoint PPT Presentation
On Hamilton-Jacobi partial differential equation and architectures of neural networks J er ome Darbon Division of Applied Mathematics, Brown University ICODE Workshop on numerical solution of HJB equation January 9, 2020 Joint work with
Division of Applied Mathematics, Brown University
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i=1) =
i∈{1,...,m}{pi, x − tθi − γi}.
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i=1) = maxi∈{1,...,m}{pi, x − tθi − γi}
i=1 are pairwise distinct, i.e., pi = pj if i = j.
i=j αiθi > θj.
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i=1) =
i∈{1,...,m}{pi, x − tθi − γi}
i=1
i∈{1,...,m}{pi, x − γi}
m
i=1 αi pi =p
i=1 αiγi
i=1),
i=1 αiθi
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i=1. Let J and H be the functions defined in Eqs. (2) and (3), respectively, and let
∂t (x, t) + H(∇xf(x, t)) = 0,
∂t (x, t) + ˜
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2
2
i=1 = {±ei}n i=1, θi = − n 2 ,
i∈{1,...,m}{pi, x − tθi − γi}, for every x ∈ Rn and t 0.
2 , 0)}m i=1.
2 , 0)}m i=1 define the following initial data and smallest Hamiltonian
2 ,
2 for every
2 for every p ∈ Bn \ {pi}m i=1.
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i∈{1,...,m}
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∂t (x, t) + ∇xH(u(x, t)) = 0,
∂t (x, t) + ∇x ˜
i=1.
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2pi2 2
2 pi 2 2−γi)/ǫ
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i=1 and let θi = − 1 2 pi2 2 for
∂t
2 ∇xfǫ(x, t)2 2 = ǫ 2△xfǫ(x, t)
i=1 e(pi ,x−γi )/ǫ
ǫ→0 ǫ>0
i∈{1,...,m}{pi, x + t
2 − γi}
2 ·2 2.
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j=1, where {(xj, tj)}N j=1 ⊂ Rn × [0, +∞), we
i=1) = 1
N
i=1) − S(xj, tj)|2.
{(pi ,θi ,γi )}m
i=1⊂Rn×R×R
i=1).
i∈{1,...,m}
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i
i
i
i
i
i
i
i
i
2ptrue i
2, for i ∈ {1, . . . , m}.
i
2ptrue i
2 and γtrue i
i
2ptrue i
2 and γtrue i
2ptrue i
2, for i ∈ {1, . . . , m}.
i∈{1,...,m}{ptrue i
i
i
j=1, where the points
j=1 are randomly sampled in Rn × [0, +∞) with respect to the standard normal
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i=1, and the sorted underlying true parameters
i
i
i
i=1. To be specific, the errors are computed as follows
i=1 pi − ptrue i
2
i=1 ptrue i
2
i=1 |θi − θtrue i
i=1 |θtrue i
i=1 |γi − γtrue i
i=1 |γtrue i
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