Universal Functional Regression with Neural Operator Flows

Regression on function spaces is typically limited to models with Gaussian process priors. We introduce the notion of universal functional regression, in which we aim to learn a prior distribution over non-Gaussian function spaces that remains mathematically tractable for functional regression. To do this, we develop Neural Operator Flows (OpFlow), an infinite-dimensional extension of normalizing flows. OpFlow is an invertible operator that maps the (potentially unknown) data function space into a Gaussian process, allowing for exact likelihood estimation of functional point evaluations. OpFlow enables robust and accurate uncertainty quantification via drawing posterior samples of the Gaussian process and subsequently mapping them into the data function space. We empirically study the performance of OpFlow on regression and generation tasks with data generated from Gaussian processes with known posterior forms and non-Gaussian processes, as well as real-world earthquake seismograms with an unknown closed-form distribution.

Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validation

We present a data-driven model for ground-motion synthesis using a Generative Adversarial Neural Operator (GANO) that combines recent advancements in machine learning and open access strong motion data sets to generate three-component acceleration time histories conditioned on moment magnitude ($M$), rupture distance ($R_{rup}$), time-average shear-wave velocity at the top $30m$ ($V_{S30}$), and tectonic environment or style of faulting. We use Neural Operators, a resolution invariant architecture that guarantees that the model training is independent of the data sampling frequency. We first present the conditional ground-motion synthesis algorithm (referred to heretofore as cGM-GANO) and discuss its advantages compared to previous work. Next, we verify the cGM-GANO framework using simulated ground motions generated with the Southern California Earthquake Center (SCEC) Broadband Platform (BBP). We lastly train cGM-GANO on a KiK-net dataset from Japan, showing that the framework can recover the magnitude, distance, and $V_{S30}$ scaling of Fourier amplitude and pseudo-spectral accelerations. We evaluate cGM-GANO through residual analysis with the empirical dataset as well as by comparison with conventional Ground Motion Models (GMMs) for selected ground motion scenarios. Results show that cGM-GANO produces consistent median scaling with the GMMs for the corresponding tectonic environments. The largest misfit is observed at short distances due to the scarcity of training data. With the exception of short distances, the aleatory variability of the response spectral ordinates is also well captured, especially for subduction events due to the adequacy of training data. Applications of the presented framework include generation of risk-targeted ground motions for site-specific engineering applications.

Generative Adversarial Neural Operators

We propose the generative adversarial neural operator (GANO), a generative model paradigm for learning probabilities on infinite-dimensional function spaces. The natural sciences and engineering are known to have many types of data that are sampled from infinite-dimensional function spaces, where classical finite-dimensional deep generative adversarial networks (GANs) may not be directly applicable. GANO generalizes the GAN framework and allows for the sampling of functions by learning push-forward operator maps in infinite-dimensional spaces. GANO consists of two main components, a generator neural operator and a discriminator neural functional. The inputs to the generator are samples of functions from a user-specified probability measure, e.g., Gaussian random field (GRF), and the generator outputs are synthetic data functions. The input to the discriminator is either a real or synthetic data function. In this work, we instantiate GANO using the Wasserstein criterion and show how the Wasserstein loss can be computed in infinite-dimensional spaces. We empirically study GANOs in controlled cases where both input and output functions are samples from GRFs and compare its performance to the finite-dimensional counterpart GAN. We empirically study the efficacy of GANO on real-world function data of volcanic activities and show its superior performance over GAN. Furthermore, we find that for the function-based data considered, GANOs are more stable to train than GANs and require less hyperparameter optimization.

U-NO: U-shaped Neural Operators

Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g. function spaces. Prior works on neural operators proposed a series of novel architectures to learn such maps and demonstrated unprecedented success in solving partial differential equations (PDEs). In this paper, we propose U-shaped Neural Operators U-NO, an architecture that allows for deeper neural operators compared to prior works. U-NOs exploit the problems structures in function predictions, demonstrate fast training, data efficiency, and robustness w.r.t hyperparameters choices. We study the performance of U-NO on PDE benchmarks, namely, Darcy's flow law and the Navier-Stokes equations. We show that U-NO results in average of 14% and 38% prediction improvement on the Darcy's flow and Navier-Stokes equations, respectively, over the state of art.

Seismic wave propagation and inversion with Neural Operators

Seismic wave propagation forms the basis for most aspects of seismological research, yet solving the wave equation is a major computational burden that inhibits the progress of research. This is exaspirated by the fact that new simulations must be performed when the velocity structure or source location is perturbed. Here, we explore a prototype framework for learning general solutions using a recently developed machine learning paradigm called Neural Operator. A trained Neural Operator can compute a solution in negligible time for any velocity structure or source location. We develop a scheme to train Neural Operators on an ensemble of simulations performed with random velocity models and source locations. As Neural Operators are grid-free, it is possible to evaluate solutions on higher resolution velocity models than trained on, providing additional computational efficiency. We illustrate the method with the 2D acoustic wave equation and demonstrate the method's applicability to seismic tomography, using reverse mode automatic differentiation to compute gradients of the wavefield with respect to the velocity structure. The developed procedure is nearly an order of magnitude faster than using conventional numerical methods for full waveform inversion.

Deep Learning-based Damage Mapping with InSAR Coherence Time Series

Satellite remote sensing is playing an increasing role in the rapid mapping of damage after natural disasters. In particular, synthetic aperture radar (SAR) can image the Earth's surface and map damage in all weather conditions, day and night. However, current SAR damage mapping methods struggle to separate damage from other changes in the Earth's surface. In this study, we propose a novel approach to damage mapping, combining deep learning with the full time history of SAR observations of an impacted region in order to detect anomalous variations in the Earth's surface properties due to a natural disaster. We quantify Earth surface change using time series of Interferometric SAR coherence, then use a recurrent neural network (RNN) as a probabilistic anomaly detector on these coherence time series. The RNN is first trained on pre-event coherence time series, and then forecasts a probability distribution of the coherence between pre- and post-event SAR images. The difference between the forecast and observed co-event coherence provides a measure of the confidence in the identification of damage. The method allows the user to choose a damage detection threshold that is customized for each location, based on the local behavior of coherence through time before the event. We apply this method to calculate estimates of damage for three earthquakes using multi-year time series of Sentinel-1 SAR acquisitions. Our approach shows good agreement with observed damage and quantitative improvement compared to using pre- to co-event coherence loss as a damage proxy.

HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networks

We introduce a scheme for probabilistic hypocenter inversion with Stein variational inference. Our approach uses a differentiable forward model in the form of a physics-informed neural network, which we train to solve the Eikonal equation. This allows for rapid approximation of the posterior by iteratively optimizing a collection of particles against a kernelized Stein discrepancy. We show that the method is well-equipped to handle highly non-convex posterior distributions, which are common in hypocentral inverse problems. A suite of experiments is performed to examine the influence of the various hyperparameters. Once trained, the method is valid for any network geometry within the study area without the need to build travel time tables. We show that the computational demands scale efficiently with the number of differential times, making it ideal for large-N sensing technologies like Distributed Acoustic Sensing.

Data-driven Accelerogram Synthesis using Deep Generative Models

Robust estimation of ground motions generated by scenario earthquakes is critical for many engineering applications. We leverage recent advances in Generative Adversarial Networks (GANs) to develop a new framework for synthesizing earthquake acceleration time histories. Our approach extends the Wasserstein GAN formulation to allow for the generation of ground-motions conditioned on a set of continuous physical variables. Our model is trained to approximate the intrinsic probability distribution of a massive set of strong-motion recordings from Japan. We show that the trained generator model can synthesize realistic 3-Component accelerograms conditioned on magnitude, distance, and $V_{s30}$. Our model captures the expected statistical features of the acceleration spectra and waveform envelopes. The output seismograms display clear P and S-wave arrivals with the appropriate energy content and relative onset timing. The synthesized Peak Ground Acceleration (PGA) estimates are also consistent with observations. We develop a set of metrics that allow us to assess the training process's stability and tune model hyperparameters. We further show that the trained generator network can interpolate to conditions where no earthquake ground motion recordings exist. Our approach allows the on-demand synthesis of accelerograms for engineering purposes.

EikoNet: Solving the Eikonal equation with Deep Neural Networks

The recent deep learning revolution has created an enormous opportunity for accelerating compute capabilities in the context of physics-based simulations. Here, we propose EikoNet, a deep learning approach to solving the Eikonal equation, which characterizes the first-arrival-time field in heterogeneous 3D velocity structures. Our grid-free approach allows for rapid determination of the travel time between any two points within a continuous 3D domain. These travel time solutions are allowed to violate the differential equation - which casts the problem as one of optimization - with the goal of finding network parameters that minimize the degree to which the equation is violated. In doing so, the method exploits the differentiability of neural networks to calculate the spatial gradients analytically, meaning the network can be trained on its own without ever needing solutions from a finite difference algorithm. EikoNet is rigorously tested on several velocity models and sampling methods to demonstrate robustness and versatility. Training and inference are highly parallelized, making the approach well-suited for GPUs. EikoNet has low memory overhead, and further avoids the need for travel-time lookup tables. The developed approach has important applications to earthquake hypocenter inversion, ray multi-pathing, and tomographic modeling, as well as to other fields beyond seismology where ray tracing is essential.

Extracting dispersion curves from ambient noise correlations using deep learning

We present a machine-learning approach to classifying the phases of surface wave dispersion curves. Standard FTAN analysis of surfaces observed on an array of receivers is converted to an image, of which, each pixel is classified as fundamental mode, first overtone, or noise. We use a convolutional neural network (U-net) architecture with a supervised learning objective and incorporate transfer learning. The training is initially performed with synthetic data to learn coarse structure, followed by fine-tuning of the network using approximately 10% of the real data based on human classification. The results show that the machine classification is nearly identical to the human picked phases. Expanding the method to process multiple images at once did not improve the performance. The developed technique will faciliate automated processing of large dispersion curve datasets.