Quantum Reservoir-Based Surrogate Modeling for Large-Amplitude Gust Responses of a Two-Dimensional Airfoil

The methodology is hybrid quantum-classical in a two-fold sense. First, the training data needed for constructing the surrogate is still generated by classical Computational Fluid Dynamics (CFD) simulations. Second, QRC itself is a hybrid quantum-classical machine learning (ML) approach, meaning that although nonlinear feature generation and memorization of recent inputs are off-loaded to a quantum reservoir, post-processing the reservoir output, in particular the regression, remains classical CPU duty (as does hyperparameter optimization). Given this categorization, details are as follows. Simulations are carried out using the DLR TAU-Code [9,10], which is a Finite-Volume-RANS solver. In a time-marching computation, the gust is fed in as external velocity excitation and the response of the airfoil is computed in terms of local pressures, as well as global forces and moment, see [11]. Here the so-called RAE2822 [12] is considered, a two-dimensional airfoil well-known in the aerodynamic community that shows flow features comparable to airfoils from transonic transport aircraft configurations. These simulations provide multivariate time series data (gust velocity and resulting lift and pitching coefficients) for different shapes of the exciting gust, e.g., for sinusoidal gusts of different amplitudes and period lengths. The flight point is kept fixed. The core supervised learning task then consists in predicting the value of the two coefficients in the next time step when given their current value as well as the current value of the exciting gust velocity as input. Adapting parameter-aware RC [13,14,15,16] to QRC, a QRC trained on time series associated with a subset of all gust configurations - belonging to a fixed gust length but varying amplitudes (or vice versa) - is able to accurately predict the evolution of the resulting coefficients when driven with unseen gust configurations. In the literature several recurrent schemes have been proposed under the umbrella of QRC, differing in particular in how and which information about past inputs is stored (classically or quantum-mechanically) and how it is transferred from one time step to the next. This work tests both the archetypal QRC of Fujii and Nakajima [8], which uses coherently evolving memory qubits, as well as quantum extreme learning machines (see, e.g., [17,18,19,20]), which possess no quantum memory but process explicitly time-delay-embedded input. Following a common choice [8,21,22,23], each of these is employed, as a default, with a transverse-field Ising Hamiltonian for the unitary reservoir evolution between consecutive inputs. Its coupling values and the evolution time are hyperparameters, the same holds for the encoding of each reservoir input (e.g., amplitude or rotational encoding). Despite these particular choices, we expect that the insights gained with these are also informative for surrogates based on other QRC variants. Besides establishing a proof-of-concept in an established benchmark, ablation studies in the model components can be carried out, concerning both quantum-specific components (e.g., choice of encoding, Hamiltonian and observ ables) as well as reservoir-generic components (e.g., selection of input features).

As mentioned above, in principle, the proposed surrogate modeling is a hybrid quantum-classical machine learning approach, with nonlinear feature generation and temporal memory provided by the quantum system, and (ridge) regression of the reservoir readout performed by the classical CPU. However, in the context of the present work, the quantum reservoir is still treated as an idealized few-qubit system undergoing unitary evolution and is thus easily simulated in a statevector simulation using the QuTiP quantum toolbox in Python [24]. (If time permits and the idealized setup produces encouraging results, the QRC-based modeling can be further probed under more difficult, yet realistic conditions including decoherence and noise effects, in particular noise channels or shot noise affecting the reservoir readout.) The model hyperparameters are optimized using optuna [25], while the (Tikhonov-)regularized linear regression, also known as ridge regression [26,27], is solved using implementations provided by scikit-learn [28].

Following reference [29], the short-term forecast quality is quantified via the fit factor Q (roughly corresponding to 1 - NRMSE), comparing the costly CFD-generated time series for lift and pitching moments to the series' obtained by the trained (and hyperparameter-optimized) quantum reservoir computer. 

So far the methodology has being probed mainly on sinusoidal gust excitations. For such gusts we find that the QRC-based surrogates indeed are able to predict gust responses in unseen in-between gust configurations with high fidelity (fit factor above 95 percent). (As this is an ongoing project, it is also planned to extend the investigation to discrete, so-called '1-cos' gusts.) 

Furthermore, following a simple divide-and-conquer strategy, we find that when attempting to cover a two-dimensional gust parameter space (amplitude and period length) with one-dimensional parameter slices along each of which a parameter-aware QRC is trained to interpolate, the difficulty of the learning problem strongly depends on the slicing direction (i.e., which parameter to fix for bundling the co-trained time series). We discuss this in light of the fixed and finite memory capacity of the reservoir and discuss how this can partly be mitigated by an adaptive sampling strategy accompanying the parameter input channel. 

Although, at least at the time of writing, the quantum statevector simulations carried out in this work are mostly agnostic to the quantum hardware platform and do not yet include potential major experimental obstacles (e.g., noise channels, shot noise), we regard this level of idealization at this initial level of exploration as acceptable, as the focus is rather on a proof-of-concept for the use case at hand and on the mentioned ablative dimensions that already arise at this stage. Moreover, several works already suggest that certain forms of (mild) noise and dissipation can even enhance the temporal processing and generalization [8,30,31,32,33], which intuitively may prevent overfit ting or enforce the fading memory property of the reservoir. This allows for some optimism that a practical demonstration under idealized conditions stays informative and constitutes a solid foundation for subsequent research.

The presented approach of a QRC-based surrogate model has been demonstrated exemplarily in this work for the gust response prediction of a two-dimensional airfoil. Typical CFD mesh sizes of current transport aircrafts are associated with an increased model size by a factor of 50 to approximately 200. If the number of CPUs that is used for the CFD simulations remains the same, the computing times required to generate the training data increase accordingly. With regard to the global coefficients, however, there are no fundamental qualitative differences in the gust responses in comparison between the two-dimensional airfoil and the three-dimensional transport aircraft, see [3]. Hence, the subsequent process of the QRC could probably remain analogous to the approaches presented here and there is no obvious reason to expect that the resource demands - either the number of qubits or the classical computational ressources (say for performing the ridge regression) - necessarily become prohibitive. However, as is also the case with non-quantum surrogates and in particular conventional RC (Echo State Networks, Next-Generation-RC), the applicability of a machine learning method can hardly conclusively be judged without explicit experimentation, even less the required model expressivity in its precise interdependency with forecast quality. Thus, a load analysis on a commercial aircraft configuration, e.g., in the preliminary design, which is based on QRC-based surrogate models, could therefore benefit from a substantial saving in computational time, although, a reliable, numerical estimate is not possible at the present time. 

Remarks: While computation time comparisons are not yet aimed for in this work, some remarks are in order. First, the overall computation time for a complete forecast along the full parameter space depends on the distribution of workload between the CFD and surrogate components (percentual split, if assuming that the CFD-computations are equally costly in all parameter points). This, in turn, couples closely to the forecast quality of the surrogate model, which is expected to decrease when provided with less training data. The trade-off here also relates to the maximum tolerable approximation error, as defined by the user (or regulator). Before diving into these considerations, a working machine learning pipeline that enables tolerable error values in the first place (ignoring computational time) must be established. Second, all-inclusive time measurements also depend on a multitude of components along the whole machine learning pipeline, most importantly on the quantum simulation method (e.g., full statevector or matrix product state simulation), but potentially also on the ridge regression solver, hyperparameter optimization or even choice of software revisions. For fair comparisons, these eventually would also need to be optimized to be benchmarked against highly optimized CFD-code. Finally, net computational times in a real hybrid quantum-classical implementation, which is the mid- to longterm goal if one is not rather investigating quantum-inspired methods, would additionally depend on the experimental specifics of the chosen quantum hardware (also via their influence on forecast quality). This falls outside the scope of the current work.