Active Manifolds: A non-linear analogue to Active Subspaces Robert - - PowerPoint PPT Presentation

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Active Manifolds: A non-linear analogue to Active Subspaces

Robert A. Bridges PhD, Oak Ridge National Laboratory bridgesra@ornl.gov Anthony Gruber, PhD, TTU Christopher Felder, MS, WUSTL Miki Verma, BS, ORNL Chelsey Hoff, BS

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Robert A. Bridges

Problem

so

• Regression:

Given recover

• Sensitivity Analysis: Which coordinate directions matter?

Can we reduce dimension of input space to make this easier?

f : Rn − → R

C1

f(x1, ..., xn) ∈ R

f

{(xi, f(xi), rf(xi))}i

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Robert A. Bridges

New Approach: Active Manifolds

Intuition: Standing at in domain of

• There are

directions one can step without changing .

• There is one, special direction,

in which changes maximally!

Level sets (orange) and gradient vector field (blue) tangent to an active manifold at every point.

f(x, y) = x3 + y3 + 0.2x + 0.6y

x0

f

n − 1

rf(x0)

f

f

Regression Algorithm Idea:

• Use gradient ascent/descent to

walk up/down hill and record values

• f —an active manifold.
• To approximate walk along a

level set to the active manifold.

f(x0)

f

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Robert A. Bridges

f1(x, y) = ey−x2

f2(x, y) = x2 + y2

Examples & Regression Results

f3(x, y) = x3 + y3 + 0.2x + 0.6y

Result: AM exhibits

• rder(s) of magnitude

less L1 and L2 average error and error variance than AS.

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Robert A. Bridges

Mathematical Foundation & Pseudo-Algorithm Presented

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Robert A. Bridges

Sensitivity Analysis for MHD Generator (Following Glaws et al. 2017)

Induced magnetic field model

0.0 0.2 0.4 0.6 0.8 1.0

Curve PDrDPeter: t

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016

CoordinDte DerivDtives of γB(t)

|log(μ)′| |log(ρ)′| |log(

dp0 dt )′|

|log(η)′| |log(B0)′|

Bind = ∂p0 ∂x lµ0 2B0 ✓ 1 − 2 √ηµ B0l tanh ✓ B0l 2√ηµ ◆◆

Parameters & ranges: Result AM allows visualization to see parameter influence throughout the active manifold:

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Robert A. Bridges

Active Manifold Benefits

• Reduces - dimensional analysis to 1 dimension

(computationally more expensive)

• Order of magnitude greater accuracy in regression over AS
• Accessible visualizations of the function and parameters

gradients along the active manifold

• Permits sensitivity analysis locally along the active manifold

Robert A. Bridges bridgesra@ornl.gov

n

Questions?

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Robert A. Bridges

Previous Approaches & Related Work

• Sliced Inverse Regression1: Given

find lower rank matrix so .

• Active Subspaces (AS) 2: Given

– Let – Do SVD on

to find directions changes most.

• ResNet Isosurface Learning3: Given

find nonlinear, lower rank (using ResNet) so .

References:

1 See Li 1991, Duan & Li 1991, Li & Naschtsheim 2006, Coudret et al. 2014 2 See Russi 2010, Constantine et al. 2014, 2015, Lukaczyk et al. 2014, Constantine & Diaz 2017 3 See Zhang & Hinkle 2019 (arxiv preprint)

{(xi, f(xi), rf(xi))}i

{(xi, f(xi))}i

B

f(x) ≈ g(Bx)

C = 1 N

N

X

i=1

rfairf T

ai = WΛWT .

C

f

{(xi, f(xi), rf(xi))}i

B

f(x) ≈ g(Bx)