SLIDE 1 Data Assimilation and Kernel Reconstruction for Nonlocal Field Dynamics
ISDA Kobe 2019
Roland Potthast
DWD & University of Reading and
Jehan Alswaihli
University of Reading
SLIDE 2
- 1. Introduction and Amari Equation
- 2. Neural State Estimation
- 3. Neural Kernel Problem
(= Deep Learning)
- 4. Integrated State and Kernel
estimation
Contents
SLIDE 3
How to use neural field models in reality?
SLIDE 4
SLIDE 5
Amari / Cowan-Wilson Equation
SLIDE 6
Amari Equation Solvability: Fixed Point Theorem
SLIDE 7
Amari Equation Example: Oscillator
SLIDE 8
Amari Equation Kernel for Oscillator
SLIDE 9
- 1. Introduction and Amari Equation
- 2. Neural State Estimation
- 3. Neural Kernel Problem
(= Deep Learning)
- 4. Integrated State and Kernel
estimation
Contents
SLIDE 10
- Consider some Pulse or Signal
- Measured at some given points (tiny electrodes)
- Or as integrated values (large electrodes)
SLIDE 11
Classical State Estimation
SLIDE 12
Covariance Matrix B
SLIDE 13
Singular Values of H for large electrode case
SLIDE 14
State Estimation Results
SLIDE 15
- 1. Introduction and Amari Equation
- 2. Neural State Estimation
- 3. Neural Kernel Problem
(= Deep Learning)
- 4. Integrated State and Kernel
estimation
Contents
SLIDE 16
SLIDE 17
A deep learning algorithm = inverse problem solution:
SLIDE 18
A deep learning algorithm = inverse problem solution:
SLIDE 19
A deep learning algorithm = inverse problem solution:
SLIDE 20
SLIDE 21
Solution with different Regularization Parameters all with strong input noise (>10%)
SLIDE 22
- 1. Introduction and Amari Equation
- 2. Neural State Estimation
- 3. Neural Kernel Problem
(= Deep Learning)
- 4. Integrated State and Kernel
estimation
Contents
SLIDE 23
Estimation and Reconstruction
SLIDE 24
Original Pulse Measurements Estimate Simulation after Rec
SLIDE 25
Est-Rec-Iteration
SLIDE 26 Convergence Result (Alswaihli and P.)
- The Transport Map is bounded
- The Estimator is convergent and bounded
- The Reconstruction is convergent and bounded
As a consequence, the iteration is convergent. For noisy data you need a stopping rule.
SLIDE 27 Original Pulse and Simulated Pulse from reconstructed Kernel
Iteration 1 Iteration 2 Iteration 4 Iteration 5
SLIDE 28 Original Pulse and Iterations from reconstructed Kernel
After 20 time steps, Iterations 1-5 After 25 time steps, Iterations 1-5
SLIDE 29
Original Pulse Simulated Pulse from learned / reconstructed Kernel (no noise)
SLIDE 30
SLIDE 31
SLIDE 32
SLIDE 33
Many Thanks!
