Eccomas 2022

# Nonlinear Model Order Reduction Schemes

In most MOR schemes, the state of $x(t) \in \mathbb R^{n}$ of a dynamical system $\begin{equation*} \dot x(t) = f(x(t)) \end{equation*}$ is encoded as $\begin{equation*} q(t) = W^Tx(t) \end{equation*}$ and decoded via $\begin{equation*} \tilde x(t) = Vq(t) \end{equation*}$ where $V$, $W\in \mathbb R^{n,r}$ are matrices.

Encoding and decoding $\begin{equation*} q(t) = W^Tx(t), \quad \tilde x(t) = Vq(t) = W^TVx(t) \end{equation*}$ with $V$, $W\in \mathbb R^{n,r}$ is a linear MOR scheme as

• $r \ll n$ – reduction of the dimension and

• $x(t)\approx \tilde x(t)=VW^Tx(t)$

## Linear MOR schemes

• fairly standard (POD, Balanced Truncation)
• fairly efficient (for linear systems or with hyperreduction like DEIM)
• inherently limited in terms of reduction versus accuracy (cp. Kolmogorov $n$-width)
• good evidence that at very low $r$, nonlinear encodings/decodings $\begin{equation*} q(t) = h(x(t)), \quad \tilde x(t) = g(q(t)) \end{equation*}$ provide better reduction vs. accuracy
• though not necessarily computational efficiency

## This talk

• Formulation of a MOR scheme with a linear quadratic encoding $\begin{equation*} \tilde x(t) = Vq(t) + \Omega \, q(t) \otimes q(t) \end{equation*}$

• use of Operator inference to identify a dynamical system $\begin{equation*} M(q(t))\,\dot q(t) = A_0 + A_1\, q(t) + A_2\,q(t) \otimes q(t) \end{equation*}$

• that best approximates given data on a $r$-dimensional manifold

• numerical proof of concept for a laminar flow problem

$x(t) \approx \tilde x(t) = Vq(t) + \Omega\,q(t)\otimes q(t)$

For a general nonlinear decoding $\begin{equation*} x(t) \approx \tilde x(t) = g(q(t)) \end{equation*}$ the dynamical system $\dot x(t) = f(x(t))$ is approximated and parametrized $\begin{equation*} \dot {\tilde x}(t) = f(\tilde x(t)) \quad \leftrightarrow \quad G(q(t)) \dot q(t) = f(g(q(t)) \end{equation*}$

where $\begin{equation*} G(q(t)) := \nabla g(q(t)) \in \mathbb R^{n,r} \end{equation*}$ is the Jacobian of $g$ at $q(t)$.

With $\begin{equation*} g(q)=Vq + \Omega\,q\otimes q, \end{equation*}$ we have $\begin{equation*} G(q)\bar q = V\bar q + \Omega\,q\otimes \bar q + \Omega\,\bar q\otimes q \end{equation*}$

and an approximation/parametrization of a linear system $\dot x(t) = Ax(t)$ as $\begin{equation*} G(q)\dot q = A_1 q + A_2\, q\otimes q \end{equation*}$ with $A_1 = AV$ and $A_2 = A\Omega$.

Since for a manifold map $g\colon \mathbb R^{r}\to \mathbb R^{n}$, the Jacobian $\nabla g(q(t)) =: G(q(t))$ has full rank,

$\begin{equation*} G(q(t))^TG(q(t))\dot q(t) = G(q(t))^TA_1 q + G(q(t))^TA_2\, q\otimes q \end{equation*}$ gives a regular differential equation in $q$,

which however comes with cubic parts $\begin{equation*} M(q)\dot q(t) = \tilde A_1 q + \tilde A_2\, q\otimes q + \tilde A_3 q\otimes q \otimes q \end{equation*}$

# Operator Inference

Using data to infer a system with a quadratic decoding

We use a POD basis $V\in \mathbb R^{n,r}$ to encode a set of snapshots $\begin{equation*} [x(t_1),\ x(t_2), \dots, x(t_N) ] \to [q(t_1),\ q(t_2), \dots, q(t_N) ] \end{equation*}$ by $q(t_i) = V^Tx(t_i) \in \mathbb R^{r}$

In a first step, we infer the quadratic correction $\Omega \in \mathbb R^{N,r^2}$ via $\begin{equation*} \sum_{i=1}^N \| x(t_i) - Vq(t_i) - \Omega \, q(t_i) \otimes q(t_i)\|^2 \to \min \end{equation*}$

Next, we differentiate the snapshots to compute $\begin{equation*} \dot x(t_i) \to \dot q(t_i) = V\dot x(t_i) \end{equation*}$ and, with the Jacobian $G(q)$ at hand, we can form the derivative along the manifold $\begin{equation*} \dot {\tilde x}(t_i) = G(q(t_i))\dot q(t_i) \end{equation*}$

Finally we can solve the quadratic operator inference problem $\begin{equation*} \sum_{i=1}^N \| M(q(t_i))\,\dot q(t_i) - A_0 - A_1\, q(t_i) - A_2\, q(t_i)\otimes q(t_i)\|^2 \to \min \end{equation*}$

for

$A_0 \in \mathbb R^{r,1}, \quad A_1\in \mathbb R^{r,r}, \quad A_2 \in \mathbb R^{r, r^2}$

that fits a quadratic system to the given snapshots.

# Numerical Example

FEM Simulation of Navier-Stokes equations $\dot v + (v\cdot \nabla) v- \frac{1}{\mathsf{Re}}\Delta v + \nabla p= f,$ $\nabla \cdot v = 0.$

• 2D laminar lid driven cavity at Re=500
• About 4000 dof in the FEM model
• 400 velocity $v$ snapshots on the [0, 4.8] time interval
• Reduced order model for the velocity of size r=5,8,12
• Extrapolation to the [4.8, 6] time interval
• Comparison with POD, DMDc, OpInf

# Conclusion

## … and Outlook

• Quadratic decoding aligns well with operator inference

• Tempting theory but no decisive numerical advantages observed

• Possible ways for improvement

• Regularization of the involved optimization problem
• Inference of higher order terms

Thank You!

## References

1.
Geelen R, Wright S, Willcox K. Operator inference for non-intrusive model reduction with nonlinear manifolds. CoRR (2022) abs/2205.02304: doi:10.48550/arXiv.2205.02304
2.
Barnett J, Farhat C. Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction. CoRR (2022) abs/2204.02462: doi:10.48550/arXiv.2204.02462
3.
Benner P, Goyal P, Heiland J, Pontes Duff I. Operator inference and physics-informed learning of low-dimensional models for incompressible flows. Electron Trans Numer Anal (2022) 56:28–51. doi:10.1553/etna_vol56s28