Data Assimilation and Kernel Reconstruction for Nonlocal Field - PowerPoint PPT Presentation

Data Assimilation and Kernel Reconstruction for Nonlocal Field Dynamics Roland Potthast DWD & University of Reading and Jehan Alswaihli University of Reading ISDA Kobe 2019

Contents 1. Introduction and Amari Equation 2. Neural State Estimation 3. Neural Kernel Problem (= Deep Learning) 4. Integrated State and Kernel estimation

How to use neural field models in reality?

Amari / Cowan-Wilson Equation

Amari Equation Solvability: Fixed Point Theorem

Amari Equation Example: Oscillator

Amari Equation Kernel for Oscillator

Contents 1. Introduction and Amari Equation 2. Neural State Estimation 3. Neural Kernel Problem (= Deep Learning) 4. Integrated State and Kernel estimation

• Consider some Pulse or Signal • Measured at some given points (tiny electrodes) • Or as integrated values (large electrodes)

Classical State Estimation

Covariance Matrix B

Singular Values of H for large electrode case

State Estimation Results

Contents 1. Introduction and Amari Equation 2. Neural State Estimation 3. Neural Kernel Problem (= Deep Learning) 4. Integrated State and Kernel estimation

A deep learning algorithm = inverse problem solution:

A deep learning algorithm = inverse problem solution:

A deep learning algorithm = inverse problem solution:

Solution with different Regularization Parameters all with strong input noise (>10%)

Contents 1. Introduction and Amari Equation 2. Neural State Estimation 3. Neural Kernel Problem (= Deep Learning) 4. Integrated State and Kernel estimation

Estimation and Reconstruction

Original Pulse Measurements Estimate Simulation after Rec

Est-Rec-Iteration

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.

Original Pulse and Simulated Pulse from reconstructed Kernel Iteration 2 Iteration 1 Iteration 5 Iteration 4

After 20 time steps, Iterations 1-5 After 25 time steps, Iterations 1-5 Original Pulse and Iterations from reconstructed Kernel

Simulated Pulse Original Pulse from learned / reconstructed Kernel (no noise)

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