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Discovering and forecasting extreme events via active learning in neural operators

A preprint version of the article is available at arXiv.


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 moot. We address each of these difficulties by combining output-weighted training schemes in Bayesian experimental design (BED) with an ensemble of deep neural operators. This model-agnostic framework pairs a BED scheme that actively selects data for quantifying extreme events with an ensemble of deep neural operators that approximate infinite-dimensional nonlinear operators. We show that not only does this framework outperform Gaussian processes, but that (1) shallow ensembles of just two members perform best; (2) extremes are uncovered regardless of the state of the initial data (that is, with or without extremes); (3) our method eliminates ‘double-descent’ phenomena; (4) the use of batches of suboptimal acquisition samples 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 a scalable artificial intelligence (AI)-assisted experimental infrastructure that can efficiently infer and pinpoint critical situations across many domains, from physical to societal systems.

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Fig. 1: Active learning of extreme events in society and nature, from pandemic spikes to rogue waves, to structural ship failures.
Fig. 2: Nailing the tail, accurate PDF and danger score convergence in 50 samples.
Fig. 3: Accelerated convergence with DNOs plus extreme acquisition functions regardless of dimensionality and parallel acquisition and shallow ensembles bring computational efficiency without performance loss.
Fig. 4: Robustness to dimensionality and initial data, with or without extremes.
Fig. 5: Efficient learning of fatigue statistics for ship design.
Fig. 6: DNOs leverage functional information for mapping to the QoI.

Data availability

All relevant data for reconstructing the results, including the LAMP dataset, are provided at dnosearch_nature_cs_data55. Additionally, all data, with exception of the LAMP data, may be computed from scratch using the code found in the dnosearch56 GitHub repository. Source data are provided with this paper.

Code availability

Code pertaining to the sequential discovery algorithm of the SIR, MMT and LAMP problems is publicly available from the GitHub repository dnosearch56. The DeepONet code framework can be found within the deepxde package on GitHub. Code pertaining to the Large Amplitude Motions Program (LAMP) v4.0.9 (May 2019) is a proprietary code developed by Leidos (formerly SAIC). Additional product information about LAMP may be found by contacting Leidos at


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We acknowledge support from DARPA grant no. HR00112290029, AFOSR MURI grant no. FA9550-21-1-0058 and ONR grants nos. N00014-20-1-2366 and N00014-21-1-2357, awarded to MIT. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. We thank V. Belenky and K. Weems from NSWC at Carderock for support regarding the LAMP code, as well as A. Blanchard for helpful and stimulating discussions around the Bayesian experimental design.

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Authors and Affiliations



E.P. and T.P.S. conceived the idea and developed the neural operator–BED framework. E.P. implemented the functional search approach for neural operators. G.E.K. contributed the DNO architecture and framework. E.P. conducted the pandemic and rogue-wave simulations and numerical experiments. S.G. conducted ship simulations and numerical experiments. All authors interpreted the results. E.P. wrote the original manuscript. All authors contributed to editing the manuscript.

Corresponding authors

Correspondence to Ethan Pickering or Themistoklis P. Sapsis.

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Nature Computational Science thanks Giovanni Dematteis, Xiang Zhou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Jie Pan, in collaboration with the Nature Computational Science team.

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Pickering, E., Guth, S., Karniadakis, G.E. et al. Discovering and forecasting extreme events via active learning in neural operators. Nat Comput Sci 2, 823–833 (2022).

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