## Publications

78 results found

Schaefer F, Magri L, Polifke W, 2022, A hybrid adjoint network model for thermoacoustic optimization, *Journal of Engineering for Gas Turbines and Power: Transactions of the ASME*, Vol: 144, Pages: 1-9, ISSN: 0742-4795

A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved while maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

Jain A, Magri L, 2022, A physical model for indirect noise in non-isentropic nozzles: transfer functions and stability, *JOURNAL OF FLUID MECHANICS*, Vol: 935, ISSN: 0022-1120

Huhn F, Magri L, 2022, Gradient-free optimization of chaotic acoustics with reservoir computing, *PHYSICAL REVIEW FLUIDS*, Vol: 7, ISSN: 2469-990X

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Racca A, Magri L, 2022, Statistical Prediction of Extreme Events from Small Datasets, *Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, Vol: 13352 LNCS, Pages: 707-713, ISSN: 0302-9743

We propose Echo State Networks (ESNs) to predict the statistics of extreme events in a turbulent flow. We train the ESNs on small datasets that lack information about the extreme events. We asses whether the networks are able to extrapolate from the small imperfect datasets and predict the heavy-tail statistics that describe the events. We find that the networks correctly predict the events and improve the statistics of the system with respect to the training data in almost all cases analysed. This opens up new possibilities for the statistical prediction of extreme events in turbulence.

Schaefer F, Magri L, Polifke W, 2021, A Hybrid Adjoint Network Model for Thermoacoustic Optimization, ASME Turbo Expo 2021

We propose a method that allows the computation of the continuous adjoint network system based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. We, thus, eliminate the need to derive the adjoint equations for every element of the network model. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. We show how the obtained adjoint system may be utilized to optimize the thermoacoustic model by determining the optimal damper setting for an annular combustor.

Doan NAK, Polifke W, Magri L, 2021, Short- and long-term prediction of a chaotic flow: A physics-constrained reservoir computing approach

We propose a physics-constrained machine learning method-based on reservoir computing- to time-accurately predict extreme events and long-term velocity statistics in a model of turbulent shear flow. The method leverages the strengths of two different approaches: empirical modelling based on reservoir computing, which it learns the chaotic dynamics from data only, and physical modelling based on conservation laws, which extrapolates the dynamics when training data becomes unavailable. We show that the combination of the two approaches is able to accurately reproduce the velocity statistics and to predict the occurrence and amplitude of extreme events in a model of self-sustaining process in turbulence. In this flow, the extreme events are abrupt transitions from turbulent to quasi-laminar states, which are deterministic phenomena that cannot be traditionally predicted because of chaos. Furthermore, the physics-constrained machine learning method is shown to be robust with respect to noise. This work opens up new possibilities for synergistically enhancing data-driven methods with physical knowledge for the time-accurate prediction of chaotic flows.

Racca A, Magri L, 2021, Automatic-differentiated Physics-Informed Echo State Network (API-ESN)

We propose the Automatic-differentiated Physics-Informed Echo State Network (API-ESN). The network is constrained by the physical equations through the reservoir’s exact time-derivative, which is computed by automatic differentiation. As compared to the original Physics-Informed Echo State Network, the accuracy of the time-derivative is increased by up to seven orders of magnitude. This increased accuracy is key in chaotic dynamical systems, where errors grows exponentially in time. The network is showcased in the reconstruction of unmeasured (hidden) states of a chaotic system. The API-ESN eliminates a source of error, which is present in existing physics-informed echo state networks, in the computation of the time-derivative. This opens up new possibilities for an accurate reconstruction of chaotic dynamical states.

Doan NAK, Polifke W, Magri L, 2021, Auto-Encoded Reservoir Computing for Turbulence Learning

We present an Auto-Encoded Reservoir-Computing (AE-RC) approach to learn the dynamics of a 2D turbulent flow. The AE-RC consists of an Autoencoder, which discovers an efficient manifold representation of the flow state, and an Echo State Network, which learns the time evolution of the flow in the manifold. The AE-RC is able to both learn the time-accurate dynamics of the flow and predict its first-order statistical moments. The AE-RC approach opens up new possibilities for the spatio-temporal prediction of turbulence with machine learning.

Racca A, Magri L, 2021, Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics

An approach to the time-accurate prediction of chaotic solutions is by learning temporal patterns from data. Echo State Networks (ESNs), which are a class of Reservoir Computing, can accurately predict the chaotic dynamics well beyond the predictability time. Existing studies, however, also showed that small changes in the hyperparameters may markedly affect the network’s performance. The aim of this paper is to assess and improve the robustness of Echo State Networks for the time-accurate prediction of chaotic solutions. The goal is three-fold. First, we investigate the robustness of routinely used validation strategies. Second, we propose the Recycle Validation, and the chaotic versions of existing validation strategies, to specifically tackle the forecasting of chaotic systems. Third, we compare Bayesian optimization with the traditional Grid Search for optimal hyperparameter selection. Numerical tests are performed on two prototypical nonlinear systems that have both chaotic and quasiperiodic solutions. Both model-free and model-informed Echo State Networks are analysed. By comparing the network’s robustness in learning chaotic versus quasiperiodic solutions, we highlight fundamental challenges in learning chaotic solutions. The proposed validation strategies, which are based on the dynamical systems properties of chaotic time series, are shown to outperform the state-of-the-art validation strategies. Because the strategies are principled-they are based on chaos theory such as the Lyapunov time-they can be applied to other Recurrent Neural Networks architectures with little modification. This work opens up new possibilities for the robust design and application of Echo State Networks, and Recurrent Neural Networks, to the time-accurate prediction of chaotic systems.

De Domenico F, Rolland EO, Rodrigues J,
et al., 2021, Compositional and entropy indirect noise generated in subsonic non-isentropic nozzles, *Journal of Fluid Mechanics*, ISSN: 0022-1120

© 2021 The Author(s),. Published by Cambridge University Press. Indirect noise generated by the acceleration of synthetic compositional and entropic perturbations through non-isentropic nozzles is measured experimentally. A physics-based analytical low-order model to evaluate the indirect noise generated by non-isentropic compact nozzles is developed and validated with experimental measurements. A one-dimensional model for describing the waves generated by the addition of mass, momentum, energy and species to a steady flow in an entropy and composition wave generator is presented. The transfer functions describing the multiple reflections of acoustic waves in an enclosed environment are derived. This analytical framework allows unambiguous identification and isolation of the experimental direct and indirect noise generated by the injection of helium, methane, argon or carbon dioxide into a flow duct. Experimental data show that entropic and compositional noise make a significant contribution to the overall pressure traces acquired in the entropy generator. Moreover, it is demonstrated that the isentropic modelling assumption is inadequate to capture the experimental behaviour, while the analytical model for non-isentropic nozzles successfully describes the direct and indirect noise transfer functions. The disregard for the compositional contribution and the unjustified use of the isentropic assumption can provide significantly inaccurate noise predictions. This work shows that compositional noise, as well as non-isentropicity in the system, should be considered in future thermoacoustic and combustion noise models.

Qadri UA, Magri L, Ihme M,
et al., 2021, Using adjoint-based optimization to enhance ignition in non-premixed jets., *Proc Math Phys Eng Sci*, Vol: 477, Pages: 20200472-20200472, ISSN: 1364-5021

Gradient-based optimization is used to reliably and optimally induce ignition in three examples of laminar non-premixed mixture configurations. Using time-integrated heat release as a cost functional, the non-convex optimization problem identified optimal energy source locations that coincide with the stoichiometric local mixture fraction surface for short optimization horizons, while for longer horizons, the hydrodynamics plays an increasingly important role and a balance between flow and chemistry features determines non-trivial optimal ignition locations. Rather than identifying a single optimal ignition location, the results of this study show that there may be several equally good ignition locations in a given flow configuration.

Doan NAK, Polifke W, Magri L, 2020, Physics-informed echo state networks, *Journal of Computational Science*, Vol: 47, ISSN: 1877-7503

© 2020 Elsevier B.V. We propose a physics-informed echo state network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional loss function during the training, which is based on the system’s governing equations. The additional loss function penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system and a truncation of the Charney–DeVore system. Compared to the conventional ESNs, the physics-informed ESNs improve the predictability horizon by about two Lyapunov times. This approach is also shown to be robust with regard to noise. The proposed framework shows the potential of using machine learning combined with prior physical knowledge to improve the time-accurate prediction of chaotic dynamical systems.

Huhn F, Magri L, 2020, Learning ergodic averages in chaotic systems, International conference on computational science, Pages: 124-132, ISSN: 0302-9743

© Springer Nature Switzerland AG 2020. We propose a physics-informed machine learning method to predict the time average of a chaotic attractor. The method is based on the hybrid echo state network (hESN). We assume that the system is ergodic, so the time average is equal to the ergodic average. Compared to conventional echo state networks (ESN) (purely data-driven), the hESN uses additional information from an incomplete, or imperfect, physical model. We evaluate the performance of the hESN and compare it to that of an ESN. This approach is demonstrated on a chaotic time-delayed thermoacoustic system, where the inclusion of a physical model significantly improves the accuracy of the prediction, reducing the relative error from 48% to 1%. This improvement is obtained at the low extra cost of solving a small number of ordinary differential equations that contain physical information. This framework shows the potential of using machine learning techniques combined with prior physical knowledge to improve the prediction of time-averaged quantities in chaotic systems.

Doan NAK, Polifke W, Magri L, 2020, Learning hidden states in a chaotic system: A physics-informed echo state network approach, International conference on computational science, Pages: 117-123, ISSN: 0302-9743

© Springer Nature Switzerland AG 2020. We extend the Physics-Informed Echo State Network (PI-ESN) framework to reconstruct the evolution of an unmeasured state (hidden state) in a chaotic system. The PI-ESN is trained by using (i) data, which contains no information on the unmeasured state, and (ii) the physical equations of a prototypical chaotic dynamical system. Non-noisy and noisy datasets are considered. First, it is shown that the PI-ESN can accurately reconstruct the unmeasured state. Second, the reconstruction is shown to be robust with respect to noisy data, which means that the PI-ESN acts as a denoiser. This paper opens up new possibilities for leveraging the synergy between physical knowledge and machine learning to enhance the reconstruction and prediction of unmeasured states in chaotic dynamical systems.

Huhn F, Magri L, 2020, Optimisation of chaotically perturbed acoustic limit cycles, *Nonlinear Dynamics*, Vol: 100, Pages: 1641-1657, ISSN: 0924-090X

© 2020, Springer Nature B.V. In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure. The objective of manufacturers is to design stable thermoacoustic configurations. In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation. In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down. First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time-averaged cost functionals exist and can be robustly calculated by a shadowing method. Third, we calculate the sensitivities of the time-averaged acoustic energy and Rayleigh index to small changes to the heat-source intensity and time delay. By embedding the sensitivities into a gradient-update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification.

Huhn F, Magri L, 2020, Stability, sensitivity and optimisation of chaotic acoustic oscillations, *JOURNAL OF FLUID MECHANICS*, Vol: 882, ISSN: 0022-1120

Magri L, Doan NAK, 2020, Physics-informed data-driven prediction of turbulent reacting flows with lyapunov analysis and sequential data assimilation, Data Analysis for Direct Numerical Simulations of Turbulent Combustion: From Equation-Based Analysis to Machine Learning, Pages: 177-196, ISBN: 9783030447175

© Springer Nature Switzerland AG 2020. High-fidelity simulations of turbulent reacting flows enable scientific understanding of the physics and engineering design of practical systems. Whereas Direct Numerical Simulation (DNS) is the most suitable numerical tool to understand the physics, under-resolved and large-eddy simulations offer a good compromise between accuracy and computational effort in the prediction of engineering flows. This compromise speeds up the computations but reduces the space-and-time accuracy of the prediction. The objective of this chapter is to (i) evaluate the predictability horizon of turbulent simulations with chaos theory, and (ii) enable the space-andtime- accurate prediction of rare and transient events using a Bayesian statistical learning approach based on data assimilation. The methods are applied to DNS of Moderate or Intense Low-oxygen Dilution (MILD) combustion. The predictability provides an estimate of the time horizon within which the occurrence of ignition kernels and deflagrative modes, which are considered here as rare and transient events, can be accurately predicted. The accurate detection of ignition kernels and their evolution towards deflagrative structures are well captured on a coarse (under-resolved) grid when data is assimilated from a costly refined DNS. Physically, such an accurate prediction is important to understand the stabilization mechanism of MILD combustion. These techniques enable the space-and-time-accurate prediction of rare and transient events in turbulent flows by combining under-resolved simulations and experimental data, for example, from engine sensors. This opens up new possibilities for on-the-fly calibration of reduced-order models for turbulent reacting flows.

Magri L, Juniper MP, Moeck JP, 2020, Sensitivity of the rayleigh criterion in thermoacoustics, *Journal of Fluid Mechanics*, Vol: 882, ISSN: 0022-1120

© Cambridge University Press 2019. Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the Rayleigh criterion. The Rayleigh criterion does not directly show how to alter a system to make it more stable. This is the objective of sensitivity analysis. Because thermoacoustic systems have many design parameters, adjoint sensitivity analysis has been proposed to obtain all the sensitivities with one extra calculation. Although adjoint sensitivity analysis can be carried out in both the time and the frequency domain, the frequency domain is more natural for a linear analysis. Perhaps surprisingly, the Rayleigh criterion has not yet been rigorously derived and comprehensively interpreted in the frequency domain. The contribution of this theoretical paper is threefold. First, the Rayleigh criterion is interpreted in the frequency domain with integral formulae for the complex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated both in the time and frequency domain, both with and without Lagrange multipliers (adjoint variables). The Lagrange multipliers are physically related to the system’s observables. Third, an adjoint Rayleigh criterion is proposed. The paper also points out that the conclusions of Juniper (Phys. Rev. Fluids, vol. 3, 2018, 110509) apply to the first variation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematical relations of this paper can be used to compute sensitivities directly from measurable quantities to enable optimal design.

Orchini A, Magri L, Silva CF,
et al., 2020, Degenerate perturbation theory in thermoacoustics: High-order sensitivities and exceptional points, *Journal of Fluid Mechanics*, ISSN: 0022-1120

© The Author(s), 2020. In this study, we connect concepts that have been recently developed in thermoacoustics, specifically (i) high-order spectral perturbation theory, (ii) symmetry-induced degenerate thermoacoustic modes, (iii) intrinsic thermoacoustic modes and (iv) exceptional points. Their connection helps gain physical insight into the behaviour of the thermoacoustic spectrum when parameters of the system are varied. First, we extend high-order adjoint-based perturbation theory of thermoacoustic modes to the degenerate case. We provide explicit formulae for the calculation of the eigenvalue corrections to any order. These formulae are valid for self-adjoint, non-self-adjoint or even non-normal systems; therefore, they can be applied to a large range of problems, including fluid dynamics. Second, by analysing the expansion coefficients of the eigenvalue corrections as a function of a parameter of interest, we accurately estimate the radius of convergence of the power series. Third, we connect the existence of a finite radius of convergence to the existence of singularities in parameter space. We identify these singularities as exceptional points, which correspond to defective thermoacoustic eigenvalues, with infinite sensitivity to infinitesimal changes in the parameters. At an exceptional point, two eigenvalues and their associated eigenvectors coalesce. Close to an exceptional point, strong veering of the eigenvalue trajectories is observed. As demonstrated in recent work, exceptional points naturally arise in thermoacoustic systems due to the interaction between modes of acoustic and intrinsic origin. The role of exceptional points in thermoacoustic systems sheds new light on the physics and sensitivity of thermoacoustic stability, which can be leveraged for passive control by small design modifications.

Yu H, Juniper MP, Magri L, 2020, A data-driven kinematic model of a ducted premixed flame, *Proceedings of the Combustion Institute*, ISSN: 1540-7489

© 2020 The Combustion Institute. Reduced-order models of flame dynamics can be used to predict and mitigate the emergence of thermoacoustic oscillations in the design of gas turbine and rocket engines. This process is hindered by the fact that these models, although often qualitatively correct, are not usually quantitatively accurate. As automated experiments and numerical simulations produce ever-increasing quantities of data, the question arises as to how this data can be assimilated into physics-informed reduced-order models in order to render these models quantitatively accurate. In this study, we develop and test a physics-based reduced-order model of a ducted premixed flame in which the model parameters are learned from high-speed videos of the flame. The experimental data is assimilated into a level-set solver using an ensemble Kalman filter. This leads to an optimally calibrated reduced-order model with quantified uncertainties, which accurately reproduces elaborate nonlinear features such as cusp formation and pinch-off. The reduced-order model continues to match the experiments after assimilation has been switched off. Further, the parameters of the model, which are extracted automatically, are shown to match the first-order behavior expected on physical grounds. This study shows how reduced-order models can be updated rapidly whenever new experimental or numerical data becomes available, without the data itself having to be stored.

Yu H, Juniper MP, Magri L, 2019, Combined state and parameter estimation in level-set methods, *Journal of Computational Physics*, Vol: 399, ISSN: 0021-9991

© 2019 Elsevier Inc. Reduced-order models based on level-set methods are widely used tools to qualitatively capture and track the nonlinear dynamics of an interface. The aim of this paper is to develop a physics-informed, data-driven, statistically rigorous learning algorithm for state and parameter estimation with level-set methods. A Bayesian approach based on data assimilation is introduced. Data assimilation is enabled by the ensemble Kalman filter and smoother, which are used in their probabilistic formulations. The level-set data assimilation framework is verified in one-dimensional and two-dimensional test cases, where state estimation, parameter estimation and uncertainty quantification are performed. The statistical performance of the proposed ensemble Kalman filter and smoother is quantified by twin experiments. In the twin experiments, the combined state and parameter estimation fully recovers the reference solution, which validates the proposed algorithm. The level-set data assimilation framework is then applied to the prediction of the nonlinear dynamics of a forced premixed flame, which exhibits the formation of sharp cusps and intricate topological changes, such as pinch-off events. The proposed physics-informed statistical learning algorithm opens up new possibilities for making reduced-order models of interfaces quantitatively predictive, any time that reference data is available.

Yu H, Jaravel T, Ihme M, et al., 2019, Data Assimilation and Optimal Calibration in Nonlinear Models of Flame Dynamics, ISSN: 0742-4795

Yu H, Jaravel T, Ihme M,
et al., 2019, Data Assimilation and Optimal Calibration in Nonlinear Models of Flame Dynamics, *Journal of Engineering for Gas Turbines and Power*, Vol: 141, ISSN: 0742-4795

<jats:title>Abstract</jats:title> <jats:p>We propose an on-the-fly statistical learning method to take a qualitative reduced-order model of the dynamics of a premixed flame and make it quantitatively accurate. This physics-informed data-driven method is based on the statistically optimal combination of (i) a reduced-order model of the dynamics of a premixed flame with a level-set method, (ii) high-quality data, which can be provided by experiments and/or high-fidelity simulations, and (iii) assimilation of the data into the reduced-order model to improve the prediction of the dynamics of the premixed flame. The reduced-order model learns the state and the parameters of the premixed flame on the fly with the ensemble Kalman filter, which is a Bayesian filter used, for example, in weather forecasting. The proposed method and algorithm are applied to two test cases with relevance to reacting flows and instabilities. First, the capabilities of the framework are demonstrated in a twin experiment, where the assimilated data are produced from the same model as that used in prediction. Second, the assimilated data are extracted from a high-fidelity reacting-flow direct numerical simulation (DNS), which provides the reference solution. The results are analyzed by using Bayesian statistics, which robustly provide the level of confidence in the calculations from the reduced-order model. The versatile method we propose enables the optimal calibration of computationally inexpensive reduced-order models in real-time when experimental data become available, for example, from gas-turbine sensors.</jats:p>

Yu H, Sengupta U, Juniper M, et al., 2019, Time-Accurate Calibration of a Thermoacoustic Model on Experimental Images of a Forced Premixed Flame

Thermoacoustic instabilities are a persistent challenge in the design of jet and rocket engines. The time-accurate calculation of thermoacoustic instabilities is challenging due to the presence of both aleatoric and epistemic uncertainties, as well as the extreme sensitivity to small changes in certain parameters. We extend our previous work (Yu et al., CTR summer program 2018; Yu et al., J. Eng. Gas Turbines Power 2019) by applying our recently published level-set data assimilation framework (Yu et al., J. Comput. Phys. 2019) to experimental images of a forced premixed flame. We force a Bunsen flame with a loudspeaker and record videos at different frequencies and amplitudes. Data assimilation provides an optimal estimate of the true state of a system, and improves the predicted shape and location of the flame. Parameter estimation uses the data to find a maximum-likelihood set of parameters for the model while simultaneously quantifying their uncertainty and identifying deficiencies in the model. We demonstrate our level-set data assimilation framework using both the ensemble Kalman filter and smoother. More generally, we take a physics-informed, reduced-order model and use statistical learning techniques to make it quantitatively accurate.

Yu H, Jaravel T, Ihme M, et al., 2019, Data assimilation and optimal calibration in nonlinear models of flame dynamics, ASME Turbo Expo

Copyright © 2019 ASME. We propose an on-the-fly statistical learning method to take a qualitative reduced-order model of the dynamics of a premixed flame and make it quantitatively accurate. This physics-informed data-driven method is based on the statistically optimal combination of (i) a reduced-order model of the dynamics of a premixed flame with a level-set method, (ii) high-quality data, which can be provided by experiments and/or high-fidelity simulations, and (iii) assimilation of the data into the reduced-order model to improve the prediction of the dynamics of the premixed flame. The reduced-order model learns the state and the parameters of the premixed flame on the fly with the ensemble Kalman filter, which is a Bayesian filter used, for example, in weather forecasting. The proposed method and algorithm are applied to two test cases with relevance to reacting flows and instabilities. First, the capabilities of the framework are demonstrated in a twin experiment, where the assimilated data is produced from the same model as that used in prediction. Second, the assimilated data is extracted from a high-fidelity reacting-flow direct numerical simulation (DNS), which provides the reference solution. The results are analyzed by using Bayesian statistics, which robustly provide the level of confidence in the calculations from the reduced-order model. The versatile method we propose enables the optimal calibration of computationally inexpensive reduced-order models in real time when experimental data becomes available, for example, from gas-turbine sensors.

Magri L, 2019, Adjoint characteristic decomposition of one-dimensional waves, *Journal of Computational Physics*, Vol: 388, Pages: 454-461, ISSN: 0021-9991

© 2019 Elsevier Inc. Adjoint methods enable the accurate calculation of the sensitivities of a quantity of interest. The sensitivity is obtained by solving the adjoint system, which can be derived by continuous or discrete adjoint strategies. In acoustic wave propagation, continuous and discrete adjoint methods have been developed to compute the eigenvalue sensitivity to design parameters and passive devices (Aguilar et al., 2017, [1]). In this short communication, it is shown that the continuous and discrete adjoint characteristic decompositions, and Riemann invariants, are connected by a similarity transformation. The results are shown in the Laplace domain. The adjoint characteristic decomposition is applied to a one-dimensional acoustic resonator, which contains a monopole source of sound. The proposed framework provides the foundation to tackle larger acoustic networks with a discrete adjoint approach, opening up new possibilities for adjoint-based design of problems that can be solved by the method of characteristics.

Mensah GA, Magri L, Orchini A,
et al., 2019, Effects of Asymmetry on Thermoacoustic Modes in Annular Combustors: A Higher-Order Perturbation Study, *Journal of Engineering for Gas Turbines and Power*, Vol: 141, ISSN: 0742-4795

Copyright © 2019 by ASME. Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice, the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally nonuniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact on the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz and linearized Euler equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function (FTF). The model equations are solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an "inclination rule" that explains the behavior of degenerate eigenvalues at first order is proven.

Magri L, 2019, Adjoint Methods as Design Tools in Thermoacoustics, *Applied Mechanics Reviews*, Vol: 71, ISSN: 0003-6900

© 2019 by ASME. In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor’s geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to ca

Silva CF, Yong KJ, Magri L, 2019, Thermoacoustic Modes of Quasi- One-Dimensional Combustors in the Region of Marginal Stability, *Journal of Engineering for Gas Turbines and Power*, Vol: 141, ISSN: 0742-4795

© 2019 by ASME. It may be generally believed that the thermoacoustic eigenfrequencies of a combustor with fully acoustically reflecting boundary conditions depend on both flame dynamics and geometry of the system. In this work, we show that there are situations where this understanding does not strictly apply. The purpose of this study is twofold. In the first part, we show that the resonance frequencies of two premixed combustors with fully acoustically reflecting boundary conditions in the region of marginal stability depend only on the parameters of the flame dynamics but do not depend on the combustor’s geometry. This is shown by means of a parametric study, where the time delay and the interaction index of the flame response are varied and the resulting complex eigenfrequency locus is shown. Assuming longitudinal acoustics and a low Mach number, a quasi-1D Helmholtz solver is utilized. The time delay and interaction index of the flame response are parametrically varied to calculate the complex eigenfrequency locus. It is found that all the eigenfrequency trajectories cross the real axis at a resonance frequency that depends only on the time delay. Such marginally stable frequencies are independent of the resonant cavity modes of the two combustors, i.e., the passive thermoacoustic modes. In the second part, we exploit the aforementioned observation to evaluate the critical flame gain required for the systems to become unstable at four eigenfrequencies located in the marginally stable region. A computationally efficient method is proposed. The key ingredient is to consider both direct and adjoint eigenvectors associated with the four eigenfrequencies. Hence, the sensitivity of the eigenfrequencies to changes in the gain at the region of marginal stability is evaluated with cheap and accurate calculations. This work contributes to the understanding of thermoacoustic stability of combustors. In the same manner, the understanding of the nature of distinc

Doan NAK, Polifke W, Magri L, 2019, Physics-Informed Echo State Networks for Chaotic Systems Forecasting, Pages: 192-198, ISBN: 9783030227463

© 2019, Springer Nature Switzerland AG. We propose a physics-informed Echo State Network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional loss function during the training of the ESNs, which penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system, where the physics-informed ESNs improve the predictability horizon by about two Lyapunov times as compared to conventional ESNs. The proposed framework shows the potential of using machine learning combined with prior physical knowledge to improve the time-accurate prediction of chaotic dynamical systems.

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