Page 372 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 372
IONAL APPROXIMATION FOR DATA-DRIVEN MODELING AND COMPLEXITY
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
problems. We show how real valued matrices can be obtained. We also show how the parame-
ters can be tuned to obtain a stable linear model.
Mixed interpolatory and inference for non-intrusive reduced order
nonlinear modelling
Charles Poussot-Vassal, charles.poussot-vassal@onera.fr
Onera, France
Based on input-output time-domain raw data collected from a complex simulator, the Mixed In-
terpolatory Inference (MII) process approach allows to infer a reduced-order linear or nonlinear
(e.g. bilinear or quadratic) time invariant dynamical model of the form
xˆ˙ = Aˆxˆ + Bˆu + ˆf (xˆ, u) , (1)
yˆ = Cˆxˆ + Dˆ u + gˆ(xˆ, u)
that accurately reproduces the underlying phenomena dictated by the raw data. In (1), xˆ(·) ∈
Rr, u(·) ∈ Rnu and yˆ(·) ∈ Rny , denote the reduced internal, input and approximated output
variables respectively. Moreover, ˆf and gˆ denote either quadratic or bilinear functions.
The approach is essentially based on the sequential combination of rational interpolation
(e.g. Pencil, Loewner, AAA) with a linear least square resolution.
With respect to intrusive methods, no prior knowledge on the operator is needed. In addition,
compared to the traditional non-intrusive operator inference approaches, the proposed rationale
alleviates the need of measuring and storing the original full-order model internal variables. It
is thus applicable to a wider range of applications than the standard intrusive and non-intrusive
methods. It is therefore very close to the identification field.
The MII is successfully applied on different numerically challenging application related to
pollutant dispersion. First (i) a large eddy simulation of a pollutants dispersion case over an
airport area, and second (ii) a flow simulation over a building, both involving multi-scale and
multi-physics dynamical phenomena.
Despite the simplicity of the resulting low complexity model, the proposed approach shows
satisfactory results to predict the pollutants plume pattern while being significantly faster to
simulate.
Dynamic neural networks and model order reduction for the simulation of
electronic circuits
Wil Schilders, w.h.a.schilders@tue.nl
TU Eindhoven, Netherlands
In recent years, there is a strong drive towards the use of artificial neural networks combined
with more traditional simulation techniques based upon physical modelling. Karniadakis and
his team at Brown University are frontrunners in this field, using so-called Physics Informed
Neural Networks (PINNs). At Philips Research, we worked on dynamic neural networks for
the simulation of electronic circuits. We were able to establish a 1-1 connection between the
specific networks used, with special neuron activation functions, and state space models. On the
one hand, this connection enabled us to predict the topology of the artificial neural network. For
example, the number of hidden layers turned out to depend on the multiplicity of eigenvalues
370
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
problems. We show how real valued matrices can be obtained. We also show how the parame-
ters can be tuned to obtain a stable linear model.
Mixed interpolatory and inference for non-intrusive reduced order
nonlinear modelling
Charles Poussot-Vassal, charles.poussot-vassal@onera.fr
Onera, France
Based on input-output time-domain raw data collected from a complex simulator, the Mixed In-
terpolatory Inference (MII) process approach allows to infer a reduced-order linear or nonlinear
(e.g. bilinear or quadratic) time invariant dynamical model of the form
xˆ˙ = Aˆxˆ + Bˆu + ˆf (xˆ, u) , (1)
yˆ = Cˆxˆ + Dˆ u + gˆ(xˆ, u)
that accurately reproduces the underlying phenomena dictated by the raw data. In (1), xˆ(·) ∈
Rr, u(·) ∈ Rnu and yˆ(·) ∈ Rny , denote the reduced internal, input and approximated output
variables respectively. Moreover, ˆf and gˆ denote either quadratic or bilinear functions.
The approach is essentially based on the sequential combination of rational interpolation
(e.g. Pencil, Loewner, AAA) with a linear least square resolution.
With respect to intrusive methods, no prior knowledge on the operator is needed. In addition,
compared to the traditional non-intrusive operator inference approaches, the proposed rationale
alleviates the need of measuring and storing the original full-order model internal variables. It
is thus applicable to a wider range of applications than the standard intrusive and non-intrusive
methods. It is therefore very close to the identification field.
The MII is successfully applied on different numerically challenging application related to
pollutant dispersion. First (i) a large eddy simulation of a pollutants dispersion case over an
airport area, and second (ii) a flow simulation over a building, both involving multi-scale and
multi-physics dynamical phenomena.
Despite the simplicity of the resulting low complexity model, the proposed approach shows
satisfactory results to predict the pollutants plume pattern while being significantly faster to
simulate.
Dynamic neural networks and model order reduction for the simulation of
electronic circuits
Wil Schilders, w.h.a.schilders@tue.nl
TU Eindhoven, Netherlands
In recent years, there is a strong drive towards the use of artificial neural networks combined
with more traditional simulation techniques based upon physical modelling. Karniadakis and
his team at Brown University are frontrunners in this field, using so-called Physics Informed
Neural Networks (PINNs). At Philips Research, we worked on dynamic neural networks for
the simulation of electronic circuits. We were able to establish a 1-1 connection between the
specific networks used, with special neuron activation functions, and state space models. On the
one hand, this connection enabled us to predict the topology of the artificial neural network. For
example, the number of hidden layers turned out to depend on the multiplicity of eigenvalues
370
