Leveraging Deep Operator Networks (DeepONet) for Acoustic Full Waveform Inversion (FWI)

Full Waveform Inversion (FWI) is an important geophysical technique considered in subsurface property prediction. It solves the inverse problem of predicting high-resolution Earth interior models from seismic data. Traditional FWI methods are computationally demanding. Inverse problems in geophysics often face challenges of non-uniqueness due to limited data, as data are often collected only on the surface. In this study, we introduce a novel methodology that leverages Deep Operator Networks (DeepONet) to attempt to improve both the efficiency and accuracy of FWI. The proposed DeepONet methodology inverts seismic waveforms for the subsurface velocity field. This approach is able to capture some key features of the subsurface velocity field. We have shown that the architecture can be applied to noisy seismic data with an accuracy that is better than some other machine learning methods. We also test our proposed method with out-of-distribution prediction for different velocity models. The proposed DeepONet shows comparable and better accuracy in some velocity models than some other machine learning methods. To improve the FWI workflow, we propose using the DeepONet output as a starting model for conventional FWI and that it may improve FWI performance. While we have only shown that DeepONet facilitates faster convergence than starting with a homogeneous velocity field, it may have some benefits compared to other approaches to constructing starting models. This integration of DeepONet into FWI may accelerate the inversion process and may also enhance its robustness and reliability.

Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacology

Physics-Informed Kolmogorov-Arnold Networks (PIKANs) are gaining attention as an effective counterpart to the original multilayer perceptron-based Physics-Informed Neural Networks (PINNs). Both representation models can address inverse problems and facilitate gray-box system identification. However, a comprehensive understanding of their performance in terms of accuracy and speed remains underexplored. In particular, we introduce a modified PIKAN architecture, tanh-cPIKAN, which is based on Chebyshev polynomials for parametrization of the univariate functions with an extra nonlinearity for enhanced performance. We then present a systematic investigation of how choices of the optimizer, representation, and training configuration influence the performance of PINNs and PIKANs in the context of systems pharmacology modeling. We benchmark a wide range of first-order, second-order, and hybrid optimizers, including various learning rate schedulers. We use the new Optax library to identify the most effective combinations for learning gray-boxes under ill-posed, non-unique, and data-sparse conditions. We examine the influence of model architecture (MLP vs. KAN), numerical precision (single vs. double), the need for warm-up phases for second-order methods, and sensitivity to the initial learning rate. We also assess the optimizer scalability for larger models and analyze the trade-offs introduced by JAX in terms of computational efficiency and numerical accuracy. Using two representative systems pharmacology case studies - a pharmacokinetics model and a chemotherapy drug-response model - we offer practical guidance on selecting optimizers and representation models/architectures for robust and efficient gray-box discovery. Our findings provide actionable insights for improving the training of physics-informed networks in biomedical applications and beyond.

Discovering Partially Known Ordinary Differential Equations: a Case Study on the Chemical Kinetics of Cellulose Degradation

The degree of polymerization (DP) is one of the methods for estimating the aging of the polymer based insulation systems, such as cellulose insulation in power components. The main degradation mechanisms in polymers are hydrolysis, pyrolysis, and oxidation. These mechanisms combined cause a reduction of the DP. However, the data availability for these types of problems is usually scarce. This study analyzes insulation aging using cellulose degradation data from power transformers. The aging problem for the cellulose immersed in mineral oil inside power transformers is modeled with ordinary differential equations (ODEs). We recover the governing equations of the degradation system using Physics-Informed Neural Networks (PINNs) and symbolic regression. We apply PINNs to discover the Arrhenius equation's unknown parameters in the Ekenstam ODE describing cellulose contamination content and the material aging process related to temperature for synthetic data and real DP values. A modification of the Ekenstam ODE is given by Emsley's system of ODEs, where the rate constant expressed by the Arrhenius equation decreases in time with the new formulation. We use PINNs and symbolic regression to recover the functional form of one of the ODEs of the system and to identify an unknown parameter.

A Neural Operator-Based Emulator for Regional Shallow Water Dynamics

Coastal regions are particularly vulnerable to the impacts of rising sea levels and extreme weather events. Accurate real-time forecasting of hydrodynamic processes in these areas is essential for infrastructure planning and climate adaptation. In this study, we present the Multiple-Input Temporal Operator Network (MITONet), a novel autoregressive neural emulator that employs dimensionality reduction to efficiently approximate high-dimensional numerical solvers for complex, nonlinear problems that are governed by time-dependent, parameterized partial differential equations. Although MITONet is applicable to a wide range of problems, we showcase its capabilities by forecasting regional tide-driven dynamics described by the two-dimensional shallow-water equations, while incorporating initial conditions, boundary conditions, and a varying domain parameter. We demonstrate MITONet's performance in a real-world application, highlighting its ability to make accurate predictions by extrapolating both in time and parametric space.

NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements

Multiphysics problems that are characterized by complex interactions among fluid dynamics, heat transfer, structural mechanics, and electromagnetics, are inherently challenging due to their coupled nature. While experimental data on certain state variables may be available, integrating these data with numerical solvers remains a significant challenge. Physics-informed neural networks (PINNs) have shown promising results in various engineering disciplines, particularly in handling noisy data and solving inverse problems. However, their effectiveness in forecasting nonlinear phenomena in multiphysics regimes is yet to be fully established. This study introduces NeuroSEM, a hybrid framework integrating PINNs with the high-fidelity Spectral Element Method (SEM) solver, Nektar++. NeuroSEM leverages strengths of both PINNs and SEM, providing robust solutions for multiphysics problems. PINNs are trained to assimilate data and model physical phenomena in specific subdomains, which are then integrated into Nektar++. We demonstrate the efficiency and accuracy of NeuroSEM for thermal convection in cavity flow and flow past a cylinder. The framework effectively handles data assimilation by addressing those subdomains and state variables where data are available. We applied NeuroSEM to the Rayleigh-B\'enard convection system, including cases with missing thermal boundary conditions. Our results indicate that NeuroSEM accurately models the physical phenomena and assimilates the data within the specified subdomains. The framework's plug-and-play nature facilitates its extension to other multiphysics or multiscale problems. Furthermore, NeuroSEM is optimized for an efficient execution on emerging integrated GPU-CPU architectures. This hybrid approach enhances the accuracy and efficiency of simulations, making it a powerful tool for tackling complex engineering challenges in various scientific domains.

A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks

Kolmogorov-Arnold Networks (KANs) were recently introduced as an alternative representation model to MLP. Herein, we employ KANs to construct physics-informed machine learning models (PIKANs) and deep operator models (DeepOKANs) for solving differential equations for forward and inverse problems. In particular, we compare them with physics-informed neural networks (PINNs) and deep operator networks (DeepONets), which are based on the standard MLP representation. We find that although the original KANs based on the B-splines parameterization lack accuracy and efficiency, modified versions based on low-order orthogonal polynomials have comparable performance to PINNs and DeepONet although they still lack robustness as they may diverge for different random seeds or higher order orthogonal polynomials. We visualize their corresponding loss landscapes and analyze their learning dynamics using information bottleneck theory. Our study follows the FAIR principles so that other researchers can use our benchmarks to further advance this emerging topic.

Rethinking materials simulations: Blending direct numerical simulations with neural operators

Direct numerical simulations (DNS) are accurate but computationally expensive for predicting materials evolution across timescales, due to the complexity of the underlying evolution equations, the nature of multiscale spatio-temporal interactions, and the need to reach long-time integration. We develop a new method that blends numerical solvers with neural operators to accelerate such simulations. This methodology is based on the integration of a community numerical solver with a U-Net neural operator, enhanced by a temporal-conditioning mechanism that enables accurate extrapolation and efficient time-to-solution predictions of the dynamics. We demonstrate the effectiveness of this framework on simulations of microstructure evolution during physical vapor deposition modeled via the phase-field method. Such simulations exhibit high spatial gradients due to the co-evolution of different material phases with simultaneous slow and fast materials dynamics. We establish accurate extrapolation of the coupled solver with up to 16.5$\times$ speed-up compared to DNS. This methodology is generalizable to a broad range of evolutionary models, from solid mechanics, to fluid dynamics, geophysics, climate, and more.

Randomized Forward Mode of Automatic Differentiation for Optimization Algorithms

Backpropagation within neural networks leverages a fundamental element of automatic differentiation, which is referred to as the reverse mode differentiation, or vector Jacobian Product (VJP) or, in the context of differential geometry, known as the pull-back process. The computation of gradient is important as update of neural network parameters is performed using gradient descent method. In this study, we present a genric randomized method, which updates the parameters of neural networks by using directional derivatives of loss functions computed efficiently by using forward mode AD or Jacobian vector Product (JVP). These JVP are computed along the random directions sampled from different probability distributions e.g., Bernoulli, Normal, Wigner, Laplace and Uniform distributions. The computation of gradient is performed during the forward pass of the neural network. We also present a rigorous analysis of the presented methods providing the rate of convergence along with the computational experiments deployed in scientific Machine learning in particular physics-informed neural networks and Deep Operator Networks.

AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification

Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework -- named AI-Aristotle -- combines eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model, and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. While the current work focuses on the performance of AI-Aristotle based on synthetic data, it can equally handle noisy experimental data and can even be used for black-box identification in just a few minutes on a laptop. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.

Tackling the Curse of Dimensionality with Physics-Informed Neural Networks

The curse-of-dimensionality (CoD) taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs as Richard Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. In this paper, we develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and samples randomly a subset of these dimensional pieces in each iteration of training PINNs. We theoretically prove the convergence guarantee and other desired properties of the proposed method. We experimentally demonstrate that the proposed method allows us to solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schr\"{o}dinger equations in thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. For instance, we solve nontrivial nonlinear PDEs (one HJB equation and one Black-Scholes equation) in 100,000 dimensions in 6 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, SDGD can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.