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In this talk, I will explore the machine learning approach to solving complex physical systems modeled by partial differential equations (PDEs). Since many PDE-solving problems can be framed as operator approximations, we will focus on operator learning. The discussion will begin by extending the universal approximation theorem to make it invariant to discretization, followed by an examination of distributed algorithms that can further improve the network flexibility to handle complex multiscale problems. To improve the network's ability to extrapolate, we will delve into multi-operator learning, particularly in designing foundation models that can address previously unseen problems. To mathematically quantify of these approximations, the talk will conclude with a discussion of neural scaling laws, focusing on the convergence of operator learning networks and the analysis of generalization error.

Gaussian process models are frequently used for continuously varying environmental quantities. Based on some set of observations, if we know the mean and covariance functions of the Gaussian process over some domain of interest, we can write down the conditional Gaussian distribution of the process at any unobserved location. In applications, the mean and/or covariance functions are generally estimated from the same observations used for prediction. Common practice is to ignore the uncertainties in the estimates of the mean and covariance functions when making predictive inferences, which is known as plug-in prediction. By studying the properties of spatial predictions in the frequency domain, we can gain useful insights into how using estimated covariance functions affect point predictions and uncertainty quantification. This approach is explored via simulations for some examples with known mean functions and simple parametric models for the covariance function that include a parameter controlling the smoothness of the process. The location of the predictand relative to the observations plays an essential role in the quality of plug-in predictions in a way that can be readily explained by thinking in the frequency domain. I will briefly consider extension of these ideas to joint prediction uncertainties at multiple unobserved locations, in particular, explaining how a bivariate posterior predictive density can deviate substantially from having elliptical contours.

Remote sensing data sets produced by NASA and other space agencies are a vast resource for the study of the Earth System and the physical processes that drive it. However, no remote sensing instrument actually observes these processes directly; instruments collect electromagnetic spectra aggregated over two-dimensional ground footprints or three-dimensional voxels (or sometimes just at a single point location). Inference on physical state based on these spectra occurs via complex, computationally intensive ground data processing infrastructures. As we transition from the Earth Observing System (EOS, circa 1999-2026) to the new Earth System Observatory (ESO, circa 2026) data volumes will explode. This presents both challenges and opportunities for scientists of all kinds. In this talk, I will specifically discuss ESO's Surface Biology and Geology (SBG) mission scheduled for launch later this decade. After describing the basics of remote sensing, I will delve into some of the important characteristics of SBG and its connection to both science and societal decision making.

The ability of spaceborne remote sensing data to address important Earth and climate science problems rests crucially on how well the underlying geophysical quantities can be inferred from these observations. Remote sensing instruments measure parts of the electromagnetic spectrum and use computational algorithms to infer the unobserved true physical states. However, the accompanying uncertainties, if they are provided at all, are usually incomplete. There are many reasons why including but not limited to unknown physics, computational artifacts and compromises, unknown uncertainties in the inputs, and more. In this talk I will discuss two approaches to uncertainty quantification for remote sensing data. The first is a practical methodology currently being implemented for NASA's Orbiting Carbon Observatory 2 and 3 missions. The method combines Monte Carlo simulation experiments with statistical modeling to approximate conditional distributions of unknown true states given point estimates produced by imperfect operational algorithms. Alternatively, the second approach explicitly leverages and accounts for spatial correlation in the underlying geophysical processes. This approach is more computationally demanding, but offers certain advantages that will be important for upcoming missions like Surface Biology and Geology (SBG) which will yield huge data volumes. I will review our approach and progress in spatial uncertainty quantification, and demonstrate with data from an SBG precursor mission called EMIT, currently on the International Space Station.

In this talk, we emphasize that patient data, including images, are not operable (clinically), but that digital twins are. Based on the former, the latter can be created. Subsequently, virtual clinical operations can be performed towards selection of optimal therapies. Digital twins are beginning to emerge in the field of medicine. We suggest that theranostic digital twins (TDTs) are amongst the most natural and feasible flavors of digitals twins. We elaborate on the importance of TDTs in a future where ‘one-size-fits-all’ therapeutic schemes will be transcended; e.g. in radiopharmaceutical therapies (RPTs). Personalized RPTs can be deployed, including optimized intervention parameters. Examples include optimization of injected radioactivities, sites of injection, injection intervals and profiles, and combination therapies. Multi-modal multi-scale images, combined with other data and aided by artificial intelligence (AI) techniques, can be utilized towards routine digital twinning of our patients, and will enable improved deliveries of RPTs and overall healthcare.

Representation theory seminar by Chengze Duan

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