Neural Chaos: A Spectral Stochastic Neural Operator

Building surrogate models with uncertainty quantification capabilities is essential for many engineering applications where randomness, such as variability in material properties, is unavoidable. Polynomial Chaos Expansion (PCE) is widely recognized as a to-go method for constructing stochastic solutions in both intrusive and non-intrusive ways. Its application becomes challenging, however, with complex or high-dimensional processes, as achieving accuracy requires higher-order polynomials, which can increase computational demands and or the risk of overfitting. Furthermore, PCE requires specialized treatments to manage random variables that are not independent, and these treatments may be problem-dependent or may fail with increasing complexity. In this work, we adopt the spectral expansion formalism used in PCE; however, we replace the classical polynomial basis functions with neural network (NN) basis functions to leverage their expressivity. To achieve this, we propose an algorithm that identifies NN-parameterized basis functions in a purely data-driven manner, without any prior assumptions about the joint distribution of the random variables involved, whether independent or dependent. The proposed algorithm identifies each NN-parameterized basis function sequentially, ensuring they are orthogonal with respect to the data distribution. The basis functions are constructed directly on the joint stochastic variables without requiring a tensor product structure. This approach may offer greater flexibility for complex stochastic models, while simplifying implementation compared to the tensor product structures typically used in PCE to handle random vectors. We demonstrate the effectiveness of the proposed scheme through several numerical examples of varying complexity and provide comparisons with classical PCE.

Comparing analytic and data-driven approaches to parameter identifiability: A power systems case study

Parameter identifiability refers to the capability of accurately inferring the parameter values of a model from its observations (data). Traditional analysis methods exploit analytical properties of the closed form model, in particular sensitivity analysis, to quantify the response of the model predictions to variations in parameters. Techniques developed to analyze data, specifically manifold learning methods, have the potential to complement, and even extend the scope of the traditional analytical approaches. We report on a study comparing and contrasting analytical and data-driven approaches to quantify parameter identifiability and, importantly, perform parameter reduction tasks. We use the infinite bus synchronous generator model, a well-understood model from the power systems domain, as our benchmark problem. Our traditional analysis methods use the Fisher Information Matrix to quantify parameter identifiability analysis, and the Manifold Boundary Approximation Method to perform parameter reduction. We compare these results to those arrived at through data-driven manifold learning schemes: Output - Diffusion Maps and Geometric Harmonics. For our test case, we find that the two suites of tools (analytical when a model is explicitly available, as well as data-driven when the model is lacking and only measurement data are available) give (correct) comparable results; these results are also in agreement with traditional analysis based on singular perturbation theory. We then discuss the prospects of using data-driven methods for such model analysis.

A Resolution Independent Neural Operator

The Deep operator network (DeepONet) is a powerful yet simple neural operator architecture that utilizes two deep neural networks to learn mappings between infinite-dimensional function spaces. This architecture is highly flexible, allowing the evaluation of the solution field at any location within the desired domain. However, it imposes a strict constraint on the input space, requiring all input functions to be discretized at the same locations; this limits its practical applications. In this work, we introduce a Resolution Independent Neural Operator (RINO) that provides a framework to make DeepONet resolution-independent, enabling it to handle input functions that are arbitrarily, but sufficiently finely, discretized. To this end, we propose a dictionary learning algorithm to adaptively learn a set of appropriate continuous basis functions, parameterized as implicit neural representations (INRs), from the input data. These basis functions are then used to project arbitrary input function data as a point cloud onto an embedding space (i.e., a vector space of finite dimensions) with dimensionality equal to the dictionary size, which can be directly used by DeepONet without any architectural changes. In particular, we utilize sinusoidal representation networks (SIRENs) as our trainable INR basis functions. We demonstrate the robustness and applicability of RINO in handling arbitrarily (but sufficiently richly) sampled input functions during both training and inference through several numerical examples.

Active search for Bifurcations

Bifurcations mark qualitative changes of long-term behavior in dynamical systems and can often signal sudden ("hard") transitions or catastrophic events (divergences). Accurately locating them is critical not just for deeper understanding of observed dynamic behavior, but also for designing efficient interventions. When the dynamical system at hand is complex, possibly noisy, and expensive to sample, standard (e.g. continuation based) numerical methods may become impractical. We propose an active learning framework, where Bayesian Optimization is leveraged to discover saddle-node or Hopf bifurcations, from a judiciously chosen small number of vector field observations. Such an approach becomes especially attractive in systems whose state x parameter space exploration is resource-limited. It also naturally provides a framework for uncertainty quantification (aleatoric and epistemic), useful in systems with inherent stochasticity.

On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them

Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation technique can subsequently be used to learn the input-output relation, from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular ``twists'' of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations of one process to entire level sets of measurements of the process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures ``now'', while the second observation process measures ``in the future'', the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators

Deep Operator Networks (DeepOnets) have revolutionized the domain of scientific machine learning for the solution of the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space of parameters and hyperparameters. This fact, along with the requirement of substantial computational resources, poses a barrier to achieving high numerical accuracy. Here, inpsired by DeepONets and to address the above challenges, we present Random Projection-based Operator Networks (RandONets): shallow networks with random projections that learn linear and nonlinear operators. The implementation of RandONets involves: (a) incorporating random bases, thus enabling the use of shallow neural networks with a single hidden layer, where the only unknowns are the output weights of the network's weighted inner product; this reduces dramatically the dimensionality of the parameter space; and, based on this, (b) using established least-squares solvers (e.g., Tikhonov regularization and preconditioned QR decomposition) that offer superior numerical approximation properties compared to other optimization techniques used in deep-learning. In this work, we prove the universal approximation accuracy of RandONets for approximating nonlinear operators and demonstrate their efficiency in approximating linear nonlinear evolution operators (right-hand-sides (RHS)) with a focus on PDEs. We show, that for this particular task, RandONets outperform, both in terms of numerical approximation accuracy and computational cost, the ``vanilla" DeepOnets.

Integrating supervised and unsupervised learning approaches to unveil critical process inputs

This study introduces a machine learning framework tailored to large-scale industrial processes characterized by a plethora of numerical and categorical inputs. The framework aims to (i) discern critical parameters influencing the output and (ii) generate accurate out-of-sample qualitative and quantitative predictions of production outcomes. Specifically, we address the pivotal question of the significance of each input in shaping the process outcome, using an industrial Chemical Vapor Deposition (CVD) process as an example. The initial objective involves merging subject matter expertise and clustering techniques exclusively on the process output, here, coating thickness measurements at various positions in the reactor. This approach identifies groups of production runs that share similar qualitative characteristics, such as film mean thickness and standard deviation. In particular, the differences of the outcomes represented by the different clusters can be attributed to differences in specific inputs, indicating that these inputs are critical for the production outcome. Leveraging this insight, we subsequently implement supervised classification and regression methods using the identified critical process inputs. The proposed methodology proves to be valuable in scenarios with a multitude of inputs and insufficient data for the direct application of deep learning techniques, providing meaningful insights into the underlying processes.

Nonlinear Manifold Learning Determines Microgel Size from Raman Spectroscopy

Polymer particle size constitutes a crucial characteristic of product quality in polymerization. Raman spectroscopy is an established and reliable process analytical technology for in-line concentration monitoring. Recent approaches and some theoretical considerations show a correlation between Raman signals and particle sizes but do not determine polymer size from Raman spectroscopic measurements accurately and reliably. With this in mind, we propose three alternative machine learning workflows to perform this task, all involving diffusion maps, a nonlinear manifold learning technique for dimensionality reduction: (i) directly from diffusion maps, (ii) alternating diffusion maps, and (iii) conformal autoencoder neural networks. We apply the workflows to a data set of Raman spectra with associated size measured via dynamic light scattering of 47 microgel (cross-linked polymer) samples in a diameter range of 208nm to 483 nm. The conformal autoencoders substantially outperform state-of-the-art methods and results for the first time in a promising prediction of polymer size from Raman spectra.

Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks

We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation map by solving a system of inhomogeneous functional equations. The performance of the proposed PINN approach is assessed via two illustrative case studies for which the observer linearizing transformation map can be derived analytically. We also perform an uncertainty quantification analysis for the proposed PINN scheme and we compare it with conventional power-series numerical implementations, which rely on the computation of a power series solution.

Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemanniann K-means and the minimization of the sample Frechet variance on the Grassmann manifold to identify "local" principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Benard convection problem