Gianluca Iaccarino

Ettore Saetta, Renato Tognaccini, Gianluca Iaccarino

A convolutional autoencoder is trained using a database of airfoil aerodynamic simulations and assessed in terms of overall accuracy and interpretability. The goal is to predict the stall and to investigate the ability of the autoencoder to distinguish between the linear and non-linear response of the airfoil pressure distribution to changes in the angle of attack. After a sensitivity analysis on the learning infrastructure, we investigate the latent space identified by the autoencoder targeting extreme compression rates, i.e. very low-dimensional reconstructions. We also propose a strategy to use the decoder to generate new synthetic airfoil geometries and aerodynamic solutions by interpolation and extrapolation in the latent representation learned by the autoencoder.

Paul G. Constantine, Zachary del Rosario, Gianluca Iaccarino

A ridge function is a function of several variables that is constant along certain directions in its domain. Using classical dimensional analysis, we show that many physical laws are ridge functions; this fact yields insight into the structure of physical laws and motivates further study into ridge functions and their properties. We also connect dimensional analysis to modern subspace-based techniques for dimension reduction, including active subspaces in deterministic approximation and sufficient dimension reduction in statistical regression.

Hillary R. Fairbanks, Lluis Jofre, Gianluca Geraci et al.

Efficiently performing predictive studies of irradiated particle-laden turbulent flows has the potential of providing significant contributions towards better understanding and optimizing, for example, concentrated solar power systems. As there are many uncertainties inherent in such flows, uncertainty quantification is fundamental to improve the predictive capabilities of the numerical simulations. For large-scale, multi-physics problems exhibiting high-dimensional uncertainty, characterizing the stochastic solution presents a significant computational challenge as many methods require a large number of high-fidelity solves. This requirement results in the need for a possibly infeasible number of simulations when a typical converged high-fidelity simulation requires intensive computational resources. To reduce the cost of quantifying high-dimensional uncertainties, we investigate the application of a non-intrusive, bi-fidelity approximation to estimate statistics of quantities of interest associated with an irradiated particle-laden turbulent flow. This method relies on exploiting the low-rank structure of the solution to accelerate the stochastic sampling and approximation processes by means of cheaper-to-run, lower fidelity representations. The application of this bi-fidelity approximation results in accurate estimates of the QoI statistics while requiring a small number of high-fidelity model evaluations.

Pranay Seshadri, Gianluca Iaccarino, Tiziano Ghisu

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies.

Paul G. Constantine, David F. Gleich, Gianluca Iaccarino

Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.

Mark Benjamin, Gianluca Iaccarino

A novel strategy for generating datasets is developed within the context of drag prediction for automotive geometries using neural networks. A primary challenge in this space is constructing a training databse of sufficient size and diversity. Our method relies on a small number of starting data points, and provides a recipe to interpolate systematically between them, generating an arbitrary number of samples at the desired quality. We test this strategy using a realistic automotive geometry, and demonstrate that convolutional neural networks perform exceedingly well at predicting drag coefficients and surface pressures. Promising results are obtained in testing extrapolation performance. Our method can be applied to other problems of aerodynamic shape optimization.

Tailin Wu, Takashi Maruyama, Long Wei et al.

Inverse design, where we seek to design input variables in order to optimize an underlying objective function, is an important problem that arises across fields such as mechanical engineering to aerospace engineering. Inverse design is typically formulated as an optimization problem, with recent works leveraging optimization across learned dynamics models. However, as models are optimized they tend to fall into adversarial modes, preventing effective sampling. We illustrate that by instead optimizing over the learned energy function captured by the diffusion model, we can avoid such adversarial examples and significantly improve design performance. We further illustrate how such a design system is compositional, enabling us to combine multiple different diffusion models representing subcomponents of our desired system to design systems with every specified component. In an N-body interaction task and a challenging 2D multi-airfoil design task, we demonstrate that by composing the learned diffusion model at test time, our method allows us to design initial states and boundary shapes that are more complex than those in the training data. Our method generalizes to more objects for N-body dataset and discovers formation flying to minimize drag in the multi-airfoil design task. Project website and code can be found at https://github.com/AI4Science-WestlakeU/cindm.

Tony Zahtila, Ettore Saetta, Murray Cutforth et al.

Accurate and predictive scale-resolving simulations of laser-ignited rocket engines are highly time-consuming because the problem includes turbulent fuel-oxidizer mixing dynamics, laser-induced energy deposition, and high-speed flame growth. This is conflated with the large design space primarily corresponding to the laser operating conditions and target location. To enable rapid exploration and uncertainty quantification, we propose a data-driven surrogate modeling approach that combines convolutional autoencoders (cAEs) with neural ordinary differential equations (neural ODEs). The present target application of an ML-based surrogate model to leading-edge multi-physics turbulence simulation is part of a paradigm shift in the deployment of surrogate models towards increasing real-world complexity. Sequentially, the cAE spatially compresses high-dimensional flow fields into a low-dimensional latent space, wherein the system's temporal dynamics are learned via neural ODEs. Once trained, the model generates fast spatiotemporal predictions from initial conditions and specified operating inputs. By learning a surrogate to replace the entirety of the time-evolving simulation, the cost of predicting an ignition trial is reduced by several orders of magnitude, allowing efficient exploration of the input parameter space. Further, as the current framework yields a spatiotemporal field prediction, appraisal of the model output's physical grounding is more tractable. This approach marks a significant step toward real-time digital twins for laser-ignited rocket combustors and represents surrogate modeling in a complex system context.

Antonello Paolino, Gabriele Nava, Fabio Di Natale et al.

Robots with multi-modal locomotion are an active research field due to their versatility in diverse environments. In this context, additional actuation can provide humanoid robots with aerial capabilities. Flying humanoid robots face challenges in modeling and control, particularly with aerodynamic forces. This paper addresses these challenges from a technological and scientific standpoint. The technological contribution includes the mechanical design of iRonCub-Mk1, a jet-powered humanoid robot, optimized for jet engine integration, and hardware modifications for wind tunnel experiments on humanoid robots for precise aerodynamic forces and surface pressure measurements. The scientific contribution offers a comprehensive approach to model and control aerodynamic forces using classical and learning techniques. Computational Fluid Dynamics (CFD) simulations calculate aerodynamic forces, validated through wind tunnel experiments on iRonCub-Mk1. An automated CFD framework expands the aerodynamic dataset, enabling the training of a Deep Neural Network and a linear regression model. These models are integrated into a simulator for designing aerodynamic-aware controllers, validated through flight simulations and balancing experiments on the iRonCub-Mk1 physical prototype.

Paul G. Constantine, Zachary del Rosario, Gianluca Iaccarino

Classical dimensional analysis has two limitations: (i) the computed dimensionless groups are not unique, and (ii) the analysis does not measure relative importance of the dimensionless groups. We propose two algorithms for estimating unique and relevant dimensionless groups assuming the experimenter can control the system's independent variables and evaluate the corresponding dependent variable; e.g., computer experiments provide such a setting. The first algorithm is based on a response surface constructed from a set of experiments. The second algorithm uses many experiments to estimate finite differences over a range of the independent variables. Both algorithms are semi-empirical because they use experimental data to complement the dimensional analysis. We derive the algorithms by combining classical semi-empirical modeling with active subspaces, which---given a probability density on the independent variables---yield unique and relevant dimensionless groups. The connection between active subspaces and dimensional analysis also reveals that all empirical models are ridge functions, which are functions that are constant along low-dimensional subspaces in its domain. We demonstrate the proposed algorithms on the well-studied example of viscous pipe flow---both turbulent and laminar cases. The results include a new set of two dimensionless groups for turbulent pipe flow that are ordered by relevance to the system; the precise notion of relevance is closely tied to the derivative based global sensitivity metric from Sobol' and Kucherenko.

Paul Constantine, Michael Emory, Johan Larsson et al.

We present a computational analysis of the reactive flow in a hypersonic scramjet engine with focus on effects of uncertainties in the operating conditions. We employ a novel methodology based on active subspaces to characterize the effects of the input uncertainty on the scramjet performance. The active subspace identifies one-dimensional structure in the map from simulation inputs to quantity of interest that allows us to reparameterize the operating conditions; instead of seven physical parameters, we can use a single derived active variable. This dimension reduction enables otherwise infeasible uncertainty quantification, considering the simulation cost of roughly 9500 CPU-hours per run. For two values of the fuel injection rate, we use a total of 68 simulations to (i) identify the parameters that contribute the most to the variation in the output quantity of interest, (ii) estimate upper and lower bounds on the quantity of interest, (iii) classify sets of operating conditions as safe or unsafe corresponding to a threshold on the output quantity of interest, and (iv) estimate a cumulative distribution function for the quantity of interest.

Akshay Mittal, Gianluca Iaccarino

Coupled partial differential equation (PDE) systems, which often represent multi-physics models, are naturally suited for modular numerical solution methods. However, several challenges yet remain in extending the benefits of modularization practices to the task of uncertainty propagation. Since the cost of each deterministic PDE solve can be usually expected to be quite significant, statistical sampling based methods like Monte-Carlo (MC) are inefficient because they do not take advantage of the mathematical structure of the problem, and suffer for poor convergence properties. On the other hand, even if each module contains a moderate number of uncertain parameters, implementing spectral methods on the combined high-dimensional parameter space can be prohibitively expensive due to the curse of dimensionality. In this work, we present a module-based and efficient intrusive spectral projection (ISP) method for uncertainty propagation. In our proposed method, each subproblem is separated and modularized via block Gauss-Seidel (BGS) techniques, such that each module only needs to tackle the local stochastic parameter space. Moreover, the computational costs are significantly mitigated by constructing reduced chaos approximations of the input data that enter each module. We demonstrate implementations of our proposed method and its computational gains over the standard ISP method using numerical examples.

Akshay Mittal, Gianluca Iaccarino

Multi-physics models governed by coupled partial differential equation (PDE) systems, are naturally suited for partitioned, or modular numerical solution strategies. Although widely used in tackling deterministic coupled models, several challenges arise in extending the benefits of modularization to uncertainty propagation. On one hand, Monte-Carlo (MC) based methods are prohibitively expensive as the cost of each deterministic PDE solve is usually quite large, while on the other hand, even if each module contains a moderate number of uncertain parameters, implementing spectral methods on the combined high-dimensional parameter space can be prohibitively expensive. In this work, we present a reduced non-intrusive spectral projection (NISP) based uncertainty propagation method which separates and modularizes the uncertainty propagation task in each subproblem using block Gauss-Seidel (BGS) techniques. The overall computational costs in the proposed method are also mitigated by constructing reduced approximations of the input data entering each module. These reduced approximations and the corresponding quadrature rules are constructed via simple linear algebra transformations. We describe these components of the proposed algorithm assuming a generalized polynomial chaos (gPC) model of the stochastic solutions. We demonstrate our proposed method and its computational gains over the standard NISP method using numerical examples.

Domenico Quagliarella, Giovanni Petrone, Gianluca Iaccarino

A framework for robust optimization under uncertainty based on the use of the generalized inverse distribution function (GIDF), also called quantile function, is here proposed. Compared to more classical approaches that rely on the usage of statistical moments as deterministic attributes that define the objectives of the optimization process, the inverse cumulative distribution function allows for the use of all the possible information available in the probabilistic domain. Furthermore, the use of a quantile based approach leads naturally to a multi-objective methodology which allows an a-posteriori selection of the candidate design based on risk/opportunity criteria defined by the designer. Finally, the error on the estimation of the objectives due to the resolution of the GIDF will be proven to be quantifiable

Paul G. Constantine, David F. Gleich, Gianluca Iaccarino

We apply polynomial approximation methods -- known in the numerical PDEs context as spectral methods -- to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residual-minimizing Galerkin method, and we show how each can be interpreted as solving a truncated infinite system of equations; the difference between the two methods lies in where the truncation occurs. Using classical theory, we derive asymptotic error estimates related to the region of analyticity of the solution, and we present a practical residual error estimate. We verify the results with two numerical examples.