Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

Parametric PDEs appear in a wide variety of problems of engineering relevance, and their repeated solution under different parametric values in the context of many-query applications represents a major computational roadblock in achieving analysis and design objectives. Perhaps one of the most challenging applications, which lies at the core of this investigation, is encountered in the context of (random) heterogeneous media in which microstructural details determine their macroscopic properties [1]. These are found in a multitude of engineering applications, such as aligned and chopped fiber composites, porous membranes, particulate composites, cellular solids, colloids, microemulsions, concrete [1]. Their microstructural properties can vary, most often randomly, at multiple length-scales [2]. Capturing this variability requires, in general, very high-dimensional representations and very fine discretizations, which in turn imply a significant cost for each solution of the governing PDEs in order to predict their response [3]. Being able to efficiently obtain accurate solutions under varying microstructures represents a core challenge that can enable the solution of various forward analysis problems such as uncertainty quantification [4, 5]. More importantly, however, it is of relevance in the context of inverse design where one attempts to identify (families of) microstructures that achieve extremal or target properties [6]. While several different tools come into play, data-driven strategies, to which our contribution belongs, have risen into prominence in recent years [7, 8] as in many cases they have produced high-throughput, forward-model surrogates which are essential for inverting the microstructure-to-property link [9].

Designing civil structures such as bridges, dams or buildings is a complex task requiring many synergies from several experts. Each is responsible for different parts of the process. This is often done in a sequential manner, e.g. the structural engineer makes a design under the assumption of certain material properties (e.g. the strength class of the concrete), and then the material engineer optimizes the material with these restrictions. This paper proposes a holistic optimization procedure, which combines the concrete mixture design and structural simulations in a joint, forward workflow that we ultimately seek to invert. In this manner, new mixtures beyond standard ranges can be considered. Any design effort should account for the presence of uncertainties which can be aleatoric or epistemic as when data is used to calibrate physical models or identify models that fill missing links in the workflow. Inverting the causal relations established poses several challenges especially when these involve physics-based models which most often than not do not provide derivatives/sensitivities or when design constraints are present. To this end, we advocate Variational Optimization, with proposed extensions and appropriately chosen heuristics to overcome the aforementioned challenges. The proposed methodology is illustrated using the design of a precast concrete beam with the objective to minimize the global warming potential while satisfying a number of constraints associated with its load-bearing capacity after 28days according to the Eurocode, the demoulding time as computed by a complex nonlinear Finite Element model, and the maximum temperature during the hydration.

Constrained optimization of the parameters of a simulator plays a crucial role in a design process. These problems become challenging when the simulator is stochastic, computationally expensive, and the parameter space is high-dimensional. One can efficiently perform optimization only by utilizing the gradient with respect to the parameters, but these gradients are unavailable in many legacy, black-box codes. We introduce the algorithm Scout-Nd (Stochastic Constrained Optimization for N dimensions) to tackle the issues mentioned earlier by efficiently estimating the gradient, reducing the noise of the gradient estimator, and applying multi-fidelity schemes to further reduce computational effort. We validate our approach on standard benchmarks, demonstrating its effectiveness in optimizing parameters highlighting better performance compared to existing methods.

We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes (RANS) simulations that incorporates aleatoric, model uncertainty. The proposed closure consists of two parts. A parametric one, which utilizes previously proposed, neural-network-based tensor basis functions dependent on the rate of strain and rotation tensor invariants. This is complemented by latent, random variables which account for aleatoric model errors. A fully Bayesian formulation is proposed, combined with a sparsity-inducing prior in order to identify regions in the problem domain where the parametric closure is insufficient and where stochastic corrections to the Reynolds stress tensor are needed. Training is performed using sparse, indirect data, such as mean velocities and pressures, in contrast to the majority of alternatives that require direct Reynolds stress data. For inference and learning, a Stochastic Variational Inference scheme is employed, which is based on Monte Carlo estimates of the pertinent objective in conjunction with the reparametrization trick. This necessitates derivatives of the output of the RANS solver, for which we developed an adjoint-based formulation. In this manner, the parametric sensitivities from the differentiable solver can be combined with the built-in, automatic differentiation capability of the neural network library in order to enable an end-to-end differentiable framework. We demonstrate the capability of the proposed model to produce accurate, probabilistic, predictive estimates for all flow quantities, even in regions where model errors are present, on a separated flow in the backward-facing step benchmark problem.

Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and differentiable map between the parametric input and the branch-net output. This allows us to construct accurate approximations of the full posterior, irrespective of the number of observations and the magnitude of the observation noise, without any need for additional forward solves nor for cumbersome, iterative sampling procedures. We demonstrate the efficacy and accuracy of the proposed methodology in the context of inverse problems for three benchmarks: an anti-derivative equation, reaction-diffusion dynamics and flow through porous media.

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data. To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved. We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation. Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.

Identifying the dynamics of physical systems requires a machine learning model that can assimilate observational data, but also incorporate the laws of physics. Neural Networks based on physical principles such as the Hamiltonian or Lagrangian NNs have recently shown promising results in generating extrapolative predictions and accurately representing the system's dynamics. We show that by additionally considering the actual energy level as a regularization term during training and thus using physical information as inductive bias, the results can be further improved. Especially in the case where only small amounts of data are available, these improvements can significantly enhance the predictive capability. We apply the proposed regularization term to a Hamiltonian Neural Network (HNN) and Constrained Hamiltonian Neural Network (CHHN) for a single and double pendulum, generate predictions under unseen initial conditions and report significant gains in predictive accuracy.

While the forward and backward modeling of the process-structure-property chain has received a lot of attention from the materials community, fewer efforts have taken into consideration uncertainties. Those arise from a multitude of sources and their quantification and integration in the inversion process are essential in meeting the materials design objectives. The first contribution of this paper is a flexible, fully probabilistic formulation of such optimization problems that accounts for the uncertainty in the process-structure and structure-property linkages and enables the identification of optimal, high-dimensional, process parameters. We employ a probabilistic, data-driven surrogate for the structure-property link which expedites computations and enables handling of non-differential objectives. We couple this with a novel active learning strategy, i.e. a self-supervised collection of data, which significantly improves accuracy while requiring small amounts of training data. We demonstrate its efficacy in optimizing the mechanical and thermal properties of two-phase, random media but envision its applicability encompasses a wide variety of microstructure-sensitive design problems.

The data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a probabilistic perspective that simultaneously identifies predictive, lower-dimensional coarse-grained (CG) variables as well as their dynamics. We make use of the expressive ability of deep neural networks in order to represent the right-hand side of the CG evolution law. Furthermore, we demonstrate how domain knowledge that is very often available in the form of physical constraints (e.g. conservation laws) can be incorporated with the novel concept of virtual observables. Such constraints, apart from leading to physically realistic predictions, can significantly reduce the requisite amount of training data which enables reducing the amount of required, computationally expensive multiscale simulations (Small Data regime). The proposed state-space model is trained using probabilistic inference tools and, in contrast to several other techniques, does not require the prescription of a fine-to-coarse (restriction) projection nor time-derivatives of the state variables. The formulation adopted is capable of quantifying the predictive uncertainty as well as of reconstructing the evolution of the full, fine-scale system which allows to select the quantities of interest a posteriori. We demonstrate the efficacy of the proposed framework in a high-dimensional system of moving particles.