Operator inference for non-polynomial systems & control - PowerPoint PPT Presentation

Operator inference for non-polynomial systems & control applications 1 Boris Kr¨ amer ICERM Workshop on Mathematics of Reduced Order Models February 17-21, 2020, Providence, RI 1 Funded by: DARPA EQUiPS program award number UTA15-001067 and Air Force Center of Excellence on Multi-Fidelity Modeling of Rocket Combustor Dynamics, award FA9550-17-1-0195 1 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Motivation: Learning ROMs for complex systems Challenges in complex applications Uncertainties in model, parameters, or both Details and access to governing equations, discretization, and solver typically unavailable when working with legacy codes ⇒ Intrusive MOR infeasible in those situations Opportunities Data is everywhere (cheaper memory, better sensors, more observations); can be used to build ROMs and/or reduce uncertainties 2 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Starting comments Dynamical systems nature of problem should not be ignored in model learning. The more we know about the model, the more we can incorporate into the learning framework: Model structure, nonlinear terms, inputs, etc. 3 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Partial Differential Equation Model Full-order model (FOM) Many complex problems are modeled with Semi-discretized numerical model of the PDE: PDEs of the form: s ( t, q ) = A ( q ) s ( t ; q ) + H ( q )( s ( t ; q ) ⊗ s ( t ; q )) ˙ ∂s ∂t = A ( s ; q )+ H ( s ; q )+ f ( t, s ; q )+ B ( u ; q ) + f ( t, s ; q ) + B ( q ) u ( t ) , State s ( t ; q ) ∈ R n Input u ( t ) Parameters q ∈ R ℓ State s ( x, t ; µ ) with x ∈ Ω ⊆ R d , d = 1 , 2 , 3 Matrices A ∈ R n × n , H ∈ R n × n 2 , B ∈ R n × m Parameters q Reduced-order model (ROM) A is a linear operator ˙ s ( t ; q ) = ˆ s ( t ; q ) + ˆ ˆ A ( q )ˆ H ( q )(ˆ s ( t ; q ) ⊗ ˆ s ( t ; q )) H is quadratic in s , the nonlinear function is f ( t, s ) and B is a linear + V ⊤ f ( t, V ˆ s ( q )) + ˆ B ( q ) u ( t ) input operator. s ( t ; q ) ∈ R n Reduced state � Parameters q ∈ R ℓ Matrices ˆ A ∈ R r × r , ˆ H ∈ R r × r 2 , ˆ B ∈ R r × m 4 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Intrusive vs. non-intrusive: Sometimes we don’t have a choice 5 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Part 1: Operator Inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms (hopefully completed soon) with Peter Benner, Pawan Goyal, Benjamin Peherstorfer, Karen Willcox 6 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Part 1: Problem setting & Goal 1. Model is known in PDE form (quadratic + non-polynomial + input): ∂s ∂t = A ( s ) + H ( s ) + f ( t, s ) + B ( u ) 2. Data available from FOM 2 : s 1 , s 2 , . . . , s k , and u 1 , u 2 , . . . , u k 3. The non-polynomial nonlinear term is such that: f ( t, s ) = [ f ( t, s 1 ) , · · · , f ( t, s n )] ⊤ Goal Leverage available information of nonlinear terms to learn a ROM: s ( t ) ⊗ ′ ˆ s ( t ) = ˆ ˙ s ( t ) + ˆ s ) + ˆ s ( t )) + V ⊤ f ( t, V ˆ ˆ A ˆ H (ˆ Bu ( t ) s ( t ) = As ( t ) + H ( s ( t ) ⊗ ′ s ( t )) + f ( t, s ) + Bu ( t ) 2 The FOM data comes from time-stepping ˙ 7 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

A short example Consider the PDE ∂ts ( x, t ) = ∂ 2 ∂ ∂x 2 s ( x, t ) + e − βt s ( x, t ) − α + b ( x ) u ( t ) with time-dependent reaction term f ( t, s ) = e − βt s ( x, t ) − α . After spatial discretization with a finite difference scheme, the system reads as ( s − α := [ s i ] − α , i = 1 , 2 , . . . , n ): s ( t ) = As ( t ) + e − βt s ( t ) − α + B u ( t ) . ˙ f ( t, s ( t )) = e − βt s ( t ) − α does not require approximation of spatial derivatives Evaluating the semi-discrete nonlinear function f ( t, s ) only requires application of f ( t, s i ) at every component of s = [ s 1 , s 2 , . . . , s n ] . s Other examples : Arrhenius reaction model exp ( γ − γ s ) ; rational functions α + s , fractional powers s α etc. 8 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Intrusive projection-based ROMs Given is the following FOM, and we would like to compute a ROM: s ( t ) = As ( t ) + H ( s ( t ) ⊗ ′ s ( t )) + f ( t, s ) + Bu ( t ) ˙ V = [ v 1 , . . . , v r ] ∈ R n × r orthonormal matrix, r ≪ n , computed, e.g., with POD. Let ˜ s be the ROM state with s i ≈ V ˜ s i Projection-based (intrusive) ROM The projection-based ROM has the form s ( t ) ⊗ ′ � s ( t ) = � ˙ s ( t ) + � s ( t )) + � s ) + � � A � H ( � f ( t, � Bu ( t ) where � s ) = V ⊤ f ( t, V � f ( t, � s ) and the reduced operators H = V ⊤ H ( V ⊗ ′ V ) ∈ R r × r 2 , A = V ⊤ AV ∈ R r × r , � � � B = V ⊤ B ∈ R r × m 9 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Non-intrusive Operator Inference: Preparing the data To learn a ROM, we build on the operator inference work from [Peherstorfer and Willcox, 2016]: 1. Start with state and input data:      s 0  ,  u ( t 0 )  . S := · · · U := u ( t 1 ) · · · u ( t k ) s 1 s k 2. Due to the specific form of the nonlinear terms, we can evaluate the nonlinear snapshot matrix:    f ( t 0 , s ( t 0 ))  . F = f ( t 1 , s ( t 1 )) · · · f ( t k , s ( t k )) 3. Compute r dominant POD basis vectors of S , resulting in V s.t. � S − VV ⊤ S � ≤ tol . � S � 4. Project the state data and nolinear snapshot data ˆ ˆ S = V ⊤ S , F = V ⊤ F . 10 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Operator inference: Solving for the operators Denote with ˙ d � s k the time derivative approximation of d t ˆ s ( t k ) , which can be computed from ˆ s using a time derivative approximation. We store the time-derivative approximations in the matrix   ˙ �  ˙ ˙ ˙  . S := � s ( t 0 ) � s ( t 1 ) · · · � s ( t k ) Operator inference for non-polynomial nonlinear system A non-intrusive ROM of the form s ( t ) ⊗ ′ ˆ s ( t ) = ˆ ˙ s ( t ) + ˆ s ( t )) + ˆ s ( t )) + V ⊤ f ( t, V ˆ ˆ A ˆ H (ˆ Bu ( t ) , can be obtained by solving the optimization problem from the above projected: � ˙ S ⊗ ′ ˆ S − ˆ ˆ − ˆ A ˆ S − ˆ BU − ˆ H (ˆ min S ) � F . F � �� � A , ˆ ˆ B , ˆ H := ˆ R 11 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Two assumptions needed for convergence analysis Assumption 1 The time stepping scheme for the FOM is convergent, i.e., i ∈{ 1 ,...T/ ∆ t } � s i − s ( t i ) � 2 → 0 max as ∆ t → 0 . Assumption 2 The derivatives approximated from projected states, ˙ d ˆ s k , converge to d t ˆ s ( t k ) as the discretization time step ∆ t → 0 , i.e., s i − d i ∈{ 1 ,...T/ ∆ t } � ˙ max ˆ d t ˆ s ( t i ) � 2 → 0 as ∆ t → 0 . 12 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Convergence of the learned ROM to the intrusive ROM Theorem [Benner/Goyal/K./Peherstorfer/Willcox, 2020] Let Assumption 1 and Assumption 2 hold and let a POD basis matrix V = [ v 1 , v 2 , . . . , v r ] ∈ R n × r be given. Let � A , � B , � H be the intrusively projected ROM S ⊗ ′ ˆ operators. Let the data matrix [ˆ S , U , ˆ S ] have full column rank. Then for every ε > 0 , there exists r ≤ n and a time step size ∆ t > 0 such that the learned operators satisfy: � ˆ A − � � ˆ B − � � ˆ H − � A � ≤ ε, B � ≤ ε, H � ≤ ε. 13 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control

Hyper-reduction to speed up nonlinear function evaluation To accelerate the evaluation of V ⊤ f ( t, V ˆ s ) in the learned ROM, hyper-reduction can be used (Barrault et al., 2004; Astrid et al., 2008; Nguyen et al., 2008, Chaturantabut & Sorensen, 2010; Carlberg et al., 2013; Drmac & Gugercin, 2016,...) We employ the discrete empirical interpolation method (DEIM) to approximate ˆ s ) ≈ ˆ s ) = V ⊤ W ( S ⊤ W ) − 1 S ⊤ f ( t, V ˆ f ( t, V ˆ f r ( t, V ˆ s ) . W is computed by taking the SVD of the nonlinear snapshot matrix F and setting W to the leading m left singular vectors of F . S is an n × m matrix obtained by selecting certain columns of the n × n identity matrix, following the DEIM algorithm. 14 / 46 Boris Kr¨ amer (University of California San Diego) Operator inference for non-polynomial systems and control