Synthetic Geology -- Structural Geology Meets Deep Learning

Visualizing the first few kilometers of the Earth's subsurface, a long-standing challenge gating a virtually inexhaustible list of important applications, is coming within reach through deep learning. Building on techniques of generative artificial intelligence applied to voxelated images, we demonstrate a method that extends surface geological data supplemented by boreholes to a three-dimensional subsurface region by training a neural network. The Earth's land area having been extensively mapped for geological features, the bottleneck of this or any related technique is the availability of data below the surface. We close this data gap in the development of subsurface deep learning by designing a synthetic data-generator process that mimics eons of geological activity such as sediment compaction, volcanic intrusion, and tectonic dynamics to produce a virtually limitless number of samples of the near lithosphere. A foundation model trained on such synthetic data is able to generate a 3D image of the subsurface from a previously unseen map of surface topography and geology, showing increasing fidelity with increasing access to borehole data, depicting such structures as layers, faults, folds, dikes, and sills. We illustrate the early promise of the combination of a synthetic lithospheric generator with a trained neural network model using generative flow matching. Ultimately, such models will be fine-tuned on data from applicable campaigns, such as mineral prospecting in a given region. Though useful in itself, a regionally fine-tuned models may be employed not as an end but as a means: as an AI-based regularizer in a more traditional inverse problem application, in which the objective function represents the mismatch of additional data with physical models with applications in resource exploration, hazard assessment, and geotechnical engineering.

Training-Free Constrained Generation With Stable Diffusion Models

Stable diffusion models represent the state-of-the-art in data synthesis across diverse domains and hold transformative potential for applications in science and engineering, e.g., by facilitating the discovery of novel solutions and simulating systems that are computationally intractable to model explicitly. However, their current utility in these fields is severely limited by an inability to enforce strict adherence to physical laws and domain-specific constraints. Without this grounding, the deployment of such models in critical applications, ranging from material science to safety-critical systems, remains impractical. This paper addresses this fundamental limitation by proposing a novel approach to integrate stable diffusion models with constrained optimization frameworks, enabling them to generate outputs that satisfy stringent physical and functional requirements. We demonstrate the effectiveness of this approach through material science experiments requiring adherence to precise morphometric properties, inverse design problems involving the generation of stress-strain responses using video generation with a simulator in the loop, and safety settings where outputs must avoid copyright infringement.

PETScML: Second-order solvers for training regression problems in Scientific Machine Learning

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for the analysis, by means of deep-learning techniques, of data produced by computational science and engineering applications. At the core of these methods is the supervised training algorithm to learn the neural network realization, a highly non-convex optimization problem that is usually solved using stochastic gradient methods. However, distinct from deep-learning practice, scientific machine-learning training problems feature a much larger volume of smooth data and better characterizations of the empirical risk functions, which make them suited for conventional solvers for unconstrained optimization. We introduce a lightweight software framework built on top of the Portable and Extensible Toolkit for Scientific computation to bridge the gap between deep-learning software and conventional solvers for unconstrained minimization. We empirically demonstrate the superior efficacy of a trust region method based on the Gauss-Newton approximation of the Hessian in improving the generalization errors arising from regression tasks when learning surrogate models for a wide range of scientific machine-learning techniques and test cases. All the conventional second-order solvers tested, including L-BFGS and inexact Newton with line-search, compare favorably, either in terms of cost or accuracy, with the adaptive first-order methods used to validate the surrogate models.