Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

Abstract

This paper presents a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.

Introduction

Model reduction is effective in reducing the computational cost of simulating complex systems but remains a challenging task when the governing physics exhibit nonlinear dynamics that are not readily amenable to low-dimensional approximations. Variable transformations combined with data-driven learning of the reduced-order model (ROM) operators have emerged as one strategy to address the challenges of nonlinear model reduction [1], [2]; however, for a large class of systems, including those that arise in reacting flow and phase-field models, the desired variable transformations lead to systems of partial differential–algebraic equations (PDAEs). This paper considers the form of the PDAEs that arise in lifting a nonlinear system to polynomial form and exploits that structure to extend the Operator Inference (OpInf) approach of [3] to these lifted PDAE systems.

As a driving application, we consider a solidification process in metal additive manufacturing. Additive manufacturing is a process during which a three-dimensional part is built via the layer-by-layer deposition of material according to its digital model. Additive manufacturing’s layer-wise process adds value by allowing for the manufacturing of components with complex geometries that are either infeasible or difficult to build by conventional manufacturing processes. However, the additive manufacturing process takes place over a wide range of length scales and time scales, making numerical simulations computationally expensive. Further, uncertainty quantification is essential since the structure and properties of the resulting components are sensitive to process parameter variations [4]. Thus, ROMs are key enablers to making control, optimization, and uncertainty quantification computationally feasible for additive manufacturing. ROMs can be used in multiscale modeling of additively manufactured parts to reduce the computational costs of part-scale simulations while maintaining the desired properties at microscale [5].

Our target problem poses several challenges for existing model reduction methods. First, the transport-dominated physics of the solidification interface result in highly localized changes in the state solution with time. Classical projection-based model reduction methods that seek approximations of the state in a linear subspace (see e.g., [6], [7], [8], [9]) require many modes to achieve accuracy in a problem such as this one, rendering the resulting ROMs inefficient. Second, the forward solidification model, a coupled system of nonlinear partial differential equations (PDEs) comprising a phase-field equation and a heat equation, has a strong nonlinear dependence on the process parameters. Classical projection-based model reduction methods that use hyper-reduction methods (such as the Empirical Interpolation Method [10] and the Discrete Empirical Interpolation Method [11]) will require many interpolation points to approximate the nonlinear terms, again rendering the resulting ROMs inefficient.

Methods based on variable transformations are becoming an effective alternative for model reduction of nonlinear systems of PDEs. These approaches draw upon the notion that the introduction of auxiliary variables (often referred to as “lifting”) can lead to a reformulation of the governing equations in a structured form. For example, [12] shows how general nonlinear ordinary differential equations (ODEs) can be written as so-called “polynomial ordinary differential systems” through the introduction of additional variables. In biology, variable transformations called “recasting” are used to transform nonlinear ODEs to the so-called S-system form, a polynomial form that is faster to solve numerically [13]. Approaches based on the Koopman operator lift a nonlinear dynamical system to an infinite-dimensional space in which the dynamics are linear [14], [15]. Ref. [16] introduced the idea of reformulating nonlinear dynamical systems in quadratic form for model reduction and showed that the number of auxiliary variables needed to lift a system to quadratic-bilinear form is linear in the number of elementary nonlinear functions in the original state equations. The work in [16] shows that a large class of nonlinear terms that appear in engineering systems (including monomial, sinusoidal, and exponential terms) may be lifted to quadratic form. Lifting has been extended to model reduction of problems governed by PDEs and shown to be a competitive alternative to hyper-reduction methods [1], [17]. Yet, for many practical applications it is neither feasible nor desirable to explicitly transform the high-fidelity PDE solver, which motivates the use of non-intrusive data-driven model reduction.

Following the definitions in [18], a non-intrusive model reduction method computes the ROM using outputs of the high-fidelity simulation but without access to the full-order operators (or to their action on a vector). This is in contrast to a black-box method, which computes the ROM without using a priori or explicit knowledge of the form of the high-fidelity problem definition or its numerical implementation. Ref. [18] notes that black-box methods are non-intrusive, but not all non-intrusive methods are necessarily black-box: a non-intrusive method can exploit knowledge of the high-fidelity problem definition and the corresponding problem structure, even though it does not access the full-order operators themselves. Gray-box methods combine some knowledge of the underlying problem with a data-driven component. Gray-box methods may or may not be non-intrusive. The advantage of non-intrusive approaches is that they compute the ROM directly from simulation data, without needing access to the high-fidelity operators. Methods such as dynamic mode decomposition (DMD) [19], [20], [21], [22], [23] and OpInf [3], [18], [24] operate on snapshot data (i.e., simulated state solutions), as do approaches that directly build surrogate models of the proper orthogonal decomposition (POD) modal coefficients [25], [26], [27], [28], [29]. Other non-intrusive methods, such as those based on the Loewner framework [30], [31], [32], [33], [34], require only compressed data (e.g., input–output measurements or transfer function measurements), and are input-invariant, in contrast to OpInf where the reduced model depends on the chosen snapshots. Recent work has recognized the advantages of non-intrusive reduction methods when it comes to exploiting the power of variable transformations as discussed above. In particular, the Lift & Learn method of [2] combines lifting of a nonlinear PDE with data-driven learning of the ROM via the OpInf method of [3], so that variable transformations are applied only to snapshot data and not to the high-fidelity PDE solver itself.

For several classes of nonlinear PDEs, the particular form of the nonlinear terms means that lifting will lead to a system of PDAEs. For example, this is the case for the Arrhenius reaction terms in the tubular reactor example of [1]. It is also the case for the nonlinear thermodynamic dependencies in the solidification model considered in this paper. It is well known that reduction of PDAEs, and, in the semi-discrete case, DAEs, is challenging and that the algebraic equations require special treatment [35], [36]. In some applications the DAEs can be reformulated as a system of differential equations and model reduction techniques are applied to the index-reduced ODEs [37], but these approaches often lead to stiff systems [38]. Another approach taken in the literature is index-aware model reduction in which the nonlinear DAE is linearized about a stationary solution, the linearized DAE is decoupled into the differential and algebraic parts, and model reduction is applied to each part individually [38], [39]. These existing DAE model reduction approaches are intrusive; here we formulate a non-intrusive data-driven approach. When the algebraic equations arise through the lifting process, they take on a particular structured form. In this paper we elicit that structured form and we exploit it to learn the resulting ROM via non-intrusive operator inference. In particular, we show that the reduction of the lifted algebraic equations can be computed explicitly using only manipulations of the low-dimensional basis vectors, while the reduction of the differential equations follows the OpInf approach of [3].

Section 2 of this paper presents the lifting of a nonlinear system of PDEs to polynomial form and discusses the form of the algebraic equations that arise. Section 3 develops the proposed non-intrusive operator inference approach for the lifted system of DAEs. Section 4 presents application of the approach to solidification of a pure metal. Finally, concluding remarks are presented in Section 5.