Pau Batlle’s research while affiliated with California Institute of Technology and other places

We address functional uncertainty quantification for ill-posed inverse problems where it is possible to evaluate a possibly rank-deficient forward model, the observation noise distribution is known, and there are known parameter constraints. We present four constraint-aware confidence intervals extending the work of Batlle et al. (2023) by making the intervals both computationally feasible and less conservative. Our approach first shrinks the potentially unbounded constraint set compact in a data-adaptive way, obtains samples of the relevant test statistic inside this set to estimate a quantile function, and then uses these computed quantities to produce the intervals. Our data-adaptive bounding approach is based on the approach by Berger and Boos (1994), and involves defining a subset of the constraint set where the true parameter exists with high probability. This probabilistic guarantee is then incorporated into the final coverage guarantee in the form of an uncertainty budget. We then propose custom sampling algorithms to efficiently sample from this subset, even when the parameter space is high-dimensional. Optimization-based interval methods formulate confidence interval computation as two endpoint optimizations, where the optimization constraints can be set to achieve different types of interval calibration while seamlessly incorporating parameter constraints. However, choosing valid optimization constraints has been elusive. We show that all four proposed intervals achieve nominal coverage for a particular functional both theoretically and in practice, with numerical examples demonstrating superior performance of our intervals over the OSB interval in terms of both coverage and expected length. In particular, we show the superior performance in a realistic unfolding simulation from high-energy physics that is severely ill-posed and involves a rank-deficient forward model.

Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.

Reservoir operations for gas extraction, fluid disposal, carbon dioxide storage, or geothermal energy production are capable of inducing seismicity. Modeling tools exist for seismicity forecasting using operational data, but the computational costs and uncertainty quantification (UQ) pose challenges. We address this issue in the context of seismicity induced by gas production from the Groningen gas field using an integrated modeling framework, which combines reservoir modeling, geomechanical modeling, and stress-based earthquake forecasting. The framework is computationally efficient thanks to a 2D finite-element reservoir model, which assumes vertical flow equilibrium, and the use of semianalytical solutions to calculate poroelastic stress changes and predict seismicity rate. The earthquake nucleation model is based on rate-and-state friction and allows for an initial strength excess so that the faults are not assumed initially critically stressed. We estimate uncertainties in the predicted number of earthquakes and magnitudes. To reduce the computational costs, we assume that the stress model is true, but our UQ algorithm is general enough that the uncertainties in reservoir and stress models could be incorporated. We explore how the selection of either a Poisson or a Gaussian likelihood influences the forecast. We also use a synthetic catalog to estimate the improved forecasting performance that would have resulted from a better seismicity detection threshold. Finally, we use tapered and nontapered Gutenberg–Richter distributions to evaluate the most probable maximum magnitude over time and account for uncertainties in its estimation. Although we did not formally account for uncertainties in the stress model, we tested several alternative stress models, and found negligible impact on the predicted temporal evolution of seismicity and forecast uncertainties. Our study shows that the proposed approach yields realistic estimates of the uncertainties of temporal seismicity and is applicable for operational forecasting or induced seismicity monitoring. It can also be used in probabilistic traffic light systems.

Reservoir operations related to natural gas extraction, fluid disposal, carbon diox- ide storage, or geothermal energy production, are capable of inducing seismicity. Mod- eling tools have been developed that allow for quantitative forecasting of seismicity based on operations data, but the computational cost of such models and the difficulty in rep- resenting various sources of uncertainties make uncertainty quantification challenging. We address this issue in the context of an integrated modeling framework, which com- bines reservoir modeling, geomechanical modeling, and stress-based earthquake forecast- ing. We use the Groningen gas field as a case example of application. The modeling frame- work is computationally efficient thanks to a 2-D finite-element reservoir model which assumes vertical flow equilibrium, and the use of semi-analytical solutions to calculate poroelastic stress changes and predict seismicity rate. The earthquake nucleation model is based on rate-and-state friction and allows for an initial strength excess so that the faults are not assumed initially critically stressed. The model parameters and their un- certainties are estimated using either a Poisson or a Gaussian likelihood. We investigate the effect of the likelihood choice on the forecast performance and we estimate uncer- tainties in the predicted number of earthquakes as well as in the expected magnitudes. We use a synthetic catalog to estimate the improved forecasting performance that would have resulted from a better seismicity detection threshold. Finally, we use tapered and non-tapered Gutenberg-Richter distributions to evaluate the most probable maximum magnitude over time and account for uncertainties in its estimation. We show that the framework yields realistic estimates of the seismicity model uncertainties and is appli- cable for operational forecasting or to design induced seismicity monitoring. It could also serve as a basis for probabilistic traffic-light systems.

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator $\mathcal{G}^\dagger\,:\, \mathcal{U}\to \mathcal{V}$ are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations $\phi(u_i), \varphi(v_i)$ of input/output functions $v_i=\mathcal{G}^\dagger(u_i)$ ($i=1,\ldots,N$), and the measurement operators $\phi\,:\, \mathcal{U}\to \mathbb{R}^n$ and $\varphi\,:\, \mathcal{V} \to \mathbb{R}^m$ are linear. Writing $\psi\,:\, \mathbb{R}^n \to \mathcal{U}$ and $\chi\,:\, \mathbb{R}^m \to \mathcal{V}$ for the optimal recovery maps associated with $\phi$ and $\varphi$, we approximate $\mathcal{G}^\dagger$ with $\bar{\mathcal{G}}=\chi \circ \bar{f} \circ \phi$ where $\bar{f}$ is an optimal recovery approximation of $f^\dagger:=\varphi \circ \mathcal{G}^\dagger \circ \psi\,:\,\mathbb{R}^n \to \mathbb{R}^m$. We show that, even when using vanilla kernels (e.g., linear or Mat\'{e}rn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.