Ching-Yao Lai

Joseph Stitt, Pratik Rathore, Madeleine Udell et al.

Full waveform inversion (FWI) is the gold standard for subsurface imaging, with applications from carbon sequestration to energy and mineral exploration to earthquake hazard assessment. Machine learning approaches to FWI need field-scale, geologically diverse, and physically realistic training data, but existing resources such as Marmousi, SEAM, and OpenFWI fall short on spatial extent, temporal extent, geological diversity, and physical realism. We address these limitations with SubsurfaceGen, a GPU-accelerated generator for 3D velocity models and seismic data. Along with SubsurfaceGen, we release a paired dataset of 4,276 2D velocity slices, 5 s wavefields, and 8 s shot gathers drawn from 42 realistic, field-scale 3D velocity models, each spanning 10 km x 10 km laterally and 6.19 km deep at 10 m resolution. The dataset spans six geological settings -- four built with SubsurfaceGen and two drawn from prior sources -- relevant for carbon sequestration and hydrocarbon exploration. We use this dataset to evaluate neural operators on wavefield prediction and encoder-decoders on end-to-end velocity inversion, holding out one geological setting for out-of-distribution testing. These experiments surface failure modes at field-scale and demonstrate how SubsurfaceGen and the associated dataset can impact ML-based FWI.

Yongji Wang, Ching-Yao Lai

Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below $O(10^{-5})$ even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision $O(10^{-16})$ of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.

Jakin Ng, Yongji Wang, Ching-Yao Lai

Deep learning frameworks have become powerful tools for approaching scientific problems such as turbulent flow, which has wide-ranging applications. In practice, however, existing scientific machine learning approaches have difficulty fitting complex, multi-scale dynamical systems to very high precision, as required in scientific contexts. We propose using the novel multistage neural network approach with a spectrum-informed initialization to learn the residue from the previous stage, utilizing the spectral biases associated with neural networks to capture high frequency features in the residue, and successfully tackle the spectral bias of neural networks. This approach allows the neural network to fit target functions to double floating-point machine precision $O(10^{-16})$.

Yongji Wang, Tristan Léger, Ching-Yao Lai et al.

Recent work introduced a robust computational framework combining embedded mathematical structures, advanced optimization, and neural network architecture, leading to the discovery of multiple unstable self-similar solutions for key fluid dynamics equations, including the Incompressible Porous Media (IPM) and 2D Boussinesq systems. While this framework confirmed the existence of these singularities, an accuracy level approaching double-float machine precision was only achieved for stable and 1st unstable solutions of the 1D Córdoba-Córdoba-Fontelos model. For highly unstable solutions characterized by extreme gradients, the accuracy remained insufficient for validation. The primary obstacle is the presence of sharp solution gradients. Those gradients tend to induce large, localized PDE residuals during training, which not only hinder convergence, but also obscure the subtle signals near the origin required to identify the correct self-similar scaling parameter lambda of the solutions. In this work, we introduce a gradient-normalized PDE residual re-weighting scheme to resolve the high-gradient challenge while amplifying the critical residual signals at the origin for lambda identification. Coupled with the multi-stage neural network architecture, the PDE residuals are reduced to the level of round-off error across a wide spectrum of unstable self-similar singularities previously discovered. Furthermore, our method enables the discovery of new highly unstable singularities, i.e. the 4th unstable solution for IPM equations and a novel family of highly unstable solitons for the Nonlinear Schrödinger equations. This results in achieving high-gradient solutions with high precision, providing an important ingredient for bridging the gap between numerical discovery and computer-assisted proofs for unstable phenomena in nonlinear PDEs.

Yongji Wang, Ching-Yao Lai, Javier Gómez-Serrano et al.

Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.