Hybrid models: is divide and conquer the winning strategy?

Bram De Jaegher

Hybrid models: is divide and conquer the winning strategy?

Bram De Jaegher

Electrodialysis

Electrodialysis

Electrodialysis fouling

Electrodialysis fouling

Previously: mechanistic model ED

Previously: neural differential equations

$$\frac{d \mathbf{R}(t)}{dt} = f(\mathbf{R}(t),\, v,\, i, \, C_e)$$

NeuralODE

Previously: neural differential equations

Experiments (APAM fouling)

A simple solution often works better

Sadly, I did not find one...

The basic ED model

Purpose: prediction of boundary layer concentration (C_b)

The basic ED model

Purpose: prediction of boundary layer concentration (C_b)

The basic ED model

Purpose: prediction of boundary layer concentration (C_b)

→ Solved using the method-of-lines (DifferentialEquations.jl)

The basic ED model

Purpose: prediction of boundary layer concentration (C_b)

The basic ED model

Purpose: prediction of boundary layer concentration (C_b)

The basic ED model

Purpose: prediction of boundary layer concentration (C_b)

Fouling model

Purpose: prediction of fouling resistance

Hybrid fouling model

Purpose: prediction of fouling resistance

Hybrid fouling model

Purpose: prediction of fouling resistance

Hybrid fouling model

Purpose: prediction of fouling resistance

Hybrid fouling model

Purpose: prediction of fouling resistance

→ Neural Differential Equation

The main challenge is dividing the cake.

The cake is a lie...

It is a loss function.

The main challenge is dividing the cake.

$$a\, f(x) + NN(x, \theta)$$

$$a\,f(x) \cdot NN(x, \theta)$$

$$ L = \sum^N_{j=1} \left( \mathbf{R}_j - \hat{\mathbf{R}}_j \right)^2 $$

→ Separate training/calibration procedure.

→ Withhold some information from the data-driven model.

Divide and Conquer strategy™

Hybrid fouling model (mechanistic part)

Purpose: predict number of collisions

Hybrid fouling model (mechanistic part)

Purpose: predict number of collisions

Hybrid fouling model (mechanistic part)

Purpose: predict number of collisions

Hybrid fouling model (mechanistic part)

Purpose: predict number of collisions

Hybrid fouling model (mechanistic part)

Purpose: predict number of collisions

→ Calibrate $\gamma$

→ For membrane filtration there are tables for $\gamma$

Hybrid fouling model (mechanistic part)

Purpose: predict number of collisions

Hybrid fouling model (data-driven part)

Hybrid fouling model (data-driven part)

The model is trained with 3 time series

Current is varied

The model is tested with 9 time series

The model is tested with 9 time series

The model is tested with 9 time series

The evolution of $\alpha$ vs. mass fouling layer

Conclusions

Hybrid models are cool! if:

The mechanistic part is pretty good
You can train both models separately
You are a bit lucky

Hybrid models: divide and conquer

Bram De Jaegher, Wim De Schepper, Arne Verliefde, Ingmar Nopens