Tomohisa Okazaki

Tomohisa Okazaki

Scientific machine learning (SciML) is an interdisciplinary research field that integrates machine learning, particularly deep learning, with physics theory to understand and predict complex natural phenomena. By incorporating physical knowledge, SciML reduces the dependency on observational data, which is often limited in the natural sciences. In this article, the fundamental concepts of SciML, its applications in seismology, and prospects are described. Specifically, two popular methods are mainly discussed: physics-informed neural networks (PINNs) and neural operators (NOs). PINNs can address both forward and inverse problems by incorporating governing laws into the loss functions. The use of PINNs is expanding into areas such as simultaneous solutions of differential equations, inference in underdetermined systems, and regularization based on physics. These research directions would broaden the scope of deep learning in natural sciences. NOs are models designed for operator learning, which deals with relationships between infinite-dimensional spaces. NOs show promise in modeling the time evolution of complex systems based on observational or simulation data. Since large amounts of data are often required, combining NOs with physics-informed learning holds significant potential. Finally, SciML is considered from a broader perspective beyond deep learning: statistical (or mathematical) frameworks that integrate observational data with physical principles to model natural phenomena. In seismology, mathematically rigorous Bayesian statistics has been developed over the past decades, whereas more flexible and scalable deep learning has only emerged recently. Both approaches can be considered as part of SciML in a broad sense. Theoretical and practical insights in both directions would advance SciML methodologies and thereby deepen our understanding of earthquake phenomena.

Ryoichiro Agata, Tomohisa Okazaki

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.

Masayoshi Someya, Taisuke Yamada, Tomohisa Okazaki

The Okada model is a widely used analytical solution for displacements and strains caused by a point or rectangular dislocation source in a 3D elastic half-space. We present OkadaTorch, a PyTorch implementation of the Okada model, where the entire code is differentiable; gradients with respect to input can be easily computed using automatic differentiation (AD). Our work consists of two components: a direct translation of the original Okada model into PyTorch, and a convenient wrapper interface for efficiently computing gradients and Hessians with respect to either observation station coordinates or fault parameters. This differentiable framework is well suited for fault parameter inversion, including gradient-based optimization, Bayesian inference, and integration with scientific machine learning (SciML) models. Our code is available here: https://github.com/msomeya1/OkadaTorch

Tomohisa Okazaki

Many physical systems are described by partial differential equations (PDEs), and solving these equations and estimating their coefficients or boundary conditions (BCs) from observational data play a crucial role in understanding the associated phenomena. Recently, a machine learning approach known as physics-informed neural network, which solves PDEs using neural networks by minimizing the sum of residuals from the PDEs, BCs, and data, has gained significant attention in the scientific community. In this study, we investigate a physics-informed linear model (PILM) that uses linear combinations of basis functions to represent solutions, thereby enabling an analytical representation of optimal solutions. The PILM was formulated and verified for illustrative forward and inverse problems including cases with uncertain BCs. Furthermore, the PILM was applied to estimate crustal strain rates using geodetic data. Specifically, physical regularization that enforces elastic equilibrium on the velocity fields was compared with mathematical regularization that imposes smoothness constraints. From a Bayesian perspective, mathematical regularization exhibited superior performance. The PILM provides an analytically solvable framework applicable to linear forward and inverse problems, underdetermined systems, and physical regularization.