Probabilistic learning for prediction and optimization of complex systems Roger Ghanem Christian Soize University of Southern California Los Angeles, CA, USA Universit´ e Paris-Est, Marne-La-Vall´ ee, France MLUQ Workshop, USC, Los Angeles, CA July 24-26 2019 Roger Ghanem Christian Soize MLUQ July 26 2019 1 / 20
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Probabilistic learning for prediction and optimization ofcomplex systems
Roger GhanemChristian Soize
University of Southern California Los Angeles, CA, USAUniversite Paris-Est, Marne-La-Vallee, France
MLUQ Workshop, USC, Los Angeles, CA July 24-26 2019
Roger Ghanem Christian Soize MLUQ July 26 2019 1 / 20
Data
We are often faced with
N data points, each of dimension n.
Roger Ghanem Christian Soize MLUQ July 26 2019 2 / 20
Data
We are often faced with
N data points, each of dimension n.
Roger Ghanem Christian Soize MLUQ July 26 2019 2 / 20
Data
X = (Q,W ,U):
GIVEN VALUES FOR THESE VARIABLES PREDICT DISTRIBUTIONS FOR THESE VARIABLES
TRAINING SET
PREDICTION SET Q:U, W:
Roger Ghanem Christian Soize MLUQ July 26 2019 3 / 20
Inference
Two general approaches
Q = f (W ,U)
orFQ,|W ,U(q)
Small Data Challenge
not enough raw data to make credible inference
need additional constraints on inference
Roger Ghanem Christian Soize MLUQ July 26 2019 4 / 20
Inference
Two general approaches
Q = f (W ,U)
orFQ,|W ,U(q)
Small Data Challenge
not enough raw data to make credible inference
need additional constraints on inference
Roger Ghanem Christian Soize MLUQ July 26 2019 4 / 20
Ito equation is constructed with KD pdf as invariant measure.
d [U(r)] = [V (r)]dr
d [V (r)] = [L([U(r)])]dr − 1
2f0[V (r)]dr +
√f0[dW (r)]
I.C. [U(0)] = [Hd ], [V (0)] = [N ] a.s.
[L([U(r)])]k` =∂
∂u`log{q(u`)}
q(u`) =1
N
N∑j=1
exp
{− 1
2s2ν
‖ηd ,j − u`‖2
}Roger Ghanem Christian Soize MLUQ July 26 2019 12 / 20
Ingredients: ISDE
Previous ISDE:
admits a unique invariant measure and a unique solution([U(r)], [V (r)]), r ∈ R+ that is a second-order diffusion stochasticprocess, which is stationary and ergodic, and such that, for all r fixed inR+, the probability density of random matrix [U(r)] is p[H]([η]).
Roger Ghanem Christian Soize MLUQ July 26 2019 13 / 20