Ethan Pickering

Ethan Pickering, Stephen Guth, George Em Karniadakis et al.

Extreme events in society and nature, such as pandemic spikes, rogue waves, or structural failures, can have catastrophic consequences. Characterizing extremes is difficult as they occur rarely, arise from seemingly benign conditions, and belong to complex and often unknown infinite-dimensional systems. Such challenges render attempts at characterizing them as moot. We address each of these difficulties by combining novel training schemes in Bayesian experimental design (BED) with an ensemble of deep neural operators (DNOs). This model-agnostic framework pairs a BED scheme that actively selects data for quantifying extreme events with an ensemble of DNOs that approximate infinite-dimensional nonlinear operators. We find that not only does this framework clearly beat Gaussian processes (GPs) but that 1) shallow ensembles of just two members perform best; 2) extremes are uncovered regardless of the state of initial data (i.e. with or without extremes); 3) our method eliminates "double-descent" phenomena; 4) the use of batches of suboptimal acquisition points compared to step-by-step global optima does not hinder BED performance; and 5) Monte Carlo acquisition outperforms standard optimizers in high-dimensions. Together these conclusions form the foundation of an AI-assisted experimental infrastructure that can efficiently infer and pinpoint critical situations across many domains, from physical to societal systems.

Ethan Pickering, Themistoklis P. Sapsis

Misleading or unnecessary data can have out-sized impacts on the health or accuracy of Machine Learning (ML) models. We present a Bayesian sequential selection method, akin to Bayesian experimental design, that identifies critically important information within a dataset, while ignoring data that is either misleading or brings unnecessary complexity to the surrogate model of choice. Our method improves sample-wise error convergence and eliminates instances where more data leads to worse performance and instabilities of the surrogate model, often termed sample-wise ``double descent''. We find these instabilities are a result of the complexity of the underlying map and linked to extreme events and heavy tails. Our approach has two key features. First, the selection algorithm dynamically couples the chosen model and data. Data is chosen based on its merits towards improving the selected model, rather than being compared strictly against other data. Second, a natural convergence of the method removes the need for dividing the data into training, testing, and validation sets. Instead, the selection metric inherently assesses testing and validation error through global statistics of the model. This ensures that key information is never wasted in testing or validation. The method is applied using both Gaussian process regression and deep neural network surrogate models.

Ethan Pickering, Themistoklis P. Sapsis

We propose two bounded comparison metrics that may be implemented to arbitrary dimensions in regression tasks. One quantifies the structure of uncertainty and the other quantifies the distribution of uncertainty. The structure metric assesses the similarity in shape and location of uncertainty with the true error, while the distribution metric quantifies the supported magnitudes between the two. We apply these metrics to Gaussian Processes (GPs), Ensemble Deep Neural Nets (DNNs), and Ensemble Deep Neural Operators (DNOs) on high-dimensional and nonlinear test cases. We find that comparing a model's uncertainty estimates with the model's squared error provides a compelling ground truth assessment. We also observe that both DNNs and DNOs, especially when compared to GPs, provide encouraging metric values in high dimensions with either sparse or plentiful data.