Eccomas 2022
Jan Heiland & P. Goyal & I. Pontes-Duff & P. Benner (MPI Magdeburg)
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
\(x(t)\approx \tilde x(t)=VW^Tx(t)\)
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 \[\begin{equation*} M(q)\dot q(t) = \tilde A_1 q + \tilde A_2\, q\otimes q + \tilde A_3 q\otimes q \otimes q \end{equation*}\]
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.
FEM Simulation of Navier-Stokes equations \[ \dot v + (v\cdot \nabla) v- \frac{1}{\mathsf{Re}}\Delta v + \nabla p= f, \] \[ \nabla \cdot v = 0. \]
Re=500dof in the FEM model[0, 4.8] time intervalr=5,8,12[4.8, 6] time intervalPOD, DMDc,
OpInfQuadratic decoding aligns well with operator inference
Tempting theory but no decisive numerical advantages observed
Possible ways for improvement
Thank You!