Machine learning (ML) is a revolutionary technology with demonstrable applications across multiple disciplines. Within the Earth science community, ML has been most visible for weather forecasting, producing forecasts that rival modern physics-based models. Given the importance of deepening our understanding and improving predictions of the Earth system on all time scales, efforts are now underway to develop forecasting models into Earth-system models (ESMs), capable of representing all components of the coupled Earth system (or their aggregated behavior) and their response to external changes. Modeling the Earth system is a much more difficult problem than weather forecasting, not least because the model must represent the alternate (e.g., future) coupled states of the system for which there are no historical observations. Given that the physical principles that enable predictions about the response of the Earth system are often not explicitly coded in these ML-based models, demonstrating the credibility of ML-based ESMs thus requires us to build evidence of their consistency with the physical system. To this end, this paper puts forward five recommendations to enhance comprehensive, standardized, and independent evaluation of ML-based ESMs to strengthen their credibility and promote their wider use.

Understanding the causal interactions in simple brain tasks, such as face detection, remains a challenging and ambiguous process for researchers. In this study, we address this issue by employing a novel causal discovery method -- Directed Acyclic Graphs via M-matrices for Acyclicity (DAGMA) -- to investigate the causal structure of the brain's face-selective network and gain deeper insights into its mechanism. Using natural movie stimuli, we extract causal network of face-selective regions and analyze how frames containing faces influence this network. Our findings reveal that the presence of faces in the stimuli have causal effect both on the number and strength of causal connections within the network. Additionally, our results highlight the crucial role of subcortical regions in satisfying causal sufficiency, emphasizing its importance in causal studies of brain. This study provides a new perspective on understanding the causal architecture of the face-selective network of the brain, motivating further research on neural causality.

This study is motivated by optimal control applications that exhibit high complexity, high uncertainty, and very limited data [Wang et al., 2024, Zheng et al., 2023, Plotkin et al., 2017, Mirasol, 2017]. In particular, all of these challenges are present in the domain of biopharmaceutical manufacturing, used for production of essential life-saving treatments for severe and chronic diseases, including cancers, autoimmune disorders, metabolic diseases, genetic disorders, and infectious diseases such as COVID-19 [Zahavi and Weiner, 2020, Teo, 2022]. Using cells as factories, biomanufacturing involves hundreds of biological, physical, and chemical factors dynamically interacting with each other at molecular, cellular, and macroscopic levels and impacting production outcomes. Due to the complexity of these mechanisms, it is quite difficult to control production safely and effectively, especially in the presence of very limited data. Digital twins have proven very useful in guiding the control of complex physical systems [Tao et al., 2018].

Computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid-structure interaction (FSI) simulations. 1D models offer a computationally efficient alternative, by simplifying the 3D Navier-Stokes equations through axisymmetric flow assumption and cross-sectional averaging. However, traditional 1D models based on finite element methods (FEM) often lack accuracy compared to 3D averaged solutions. This study introduces a novel physics-constrained machine learning technique that enhances the accuracy of 1D blood flow models while maintaining computational efficiency. Our approach, utilizing a physics-constrained coupled neural differential equation (PCNDE) framework, demonstrates superior performance compared to conventional FEM-based 1D models across a wide range of inlet boundary condition waveforms and stenosis blockage ratios. A key innovation lies in the spatial formulation of the momentum conservation equation, departing from the traditional temporal approach and capitalizing on the inherent temporal periodicity of blood flow. This spatial neural differential equation formulation switches space and time and overcomes issues related to coupling stability and smoothness, while simplifying boundary condition implementation. The model accurately captures flow rate, area, and pressure variations for unseen waveforms and geometries. We evaluate the model's robustness to input noise and explore the loss landscapes associated with the inclusion of different physics terms. This advanced 1D modeling technique offers promising potential for rapid cardiovascular simulations, achieving computational efficiency and accuracy. By combining the strengths of physics-based and data-driven modeling, this approach enables fast and accurate cardiovascular simulations.

Flight test analysis often requires predefined test points with arbitrarily tight tolerances, leading to extensive and resource-intensive experimental campaigns. To address this challenge, we propose a novel approach to flight test analysis using Gaussian processes (GPs) with physics-informed mean functions to estimate aerodynamic quantities from arbitrary flight test data, validated using real T-38 aircraft data collected in collaboration with the United States Air Force Test Pilot School. We demonstrate our method by estimating the pitching moment coefficient without requiring predefined or repeated flight test points, significantly reducing the need for extensive experimental campaigns. Our approach incorporates aerodynamic models as priors within the GP framework, enhancing predictive accuracy across diverse flight conditions and providing robust uncertainty quantification. Key contributions include the integration of physics-based priors in a probabilistic model, which allows for precise computation from arbitrary flight test maneuvers, and the demonstration of our method capturing relevant dynamic characteristics such as short-period mode behavior. The proposed framework offers a scalable and generalizable solution for efficient data-driven flight test analysis and is able to accurately predict the short period frequency and damping for the T-38 across several Mach and dynamic pressure profiles.

Since its modern inception in the pioneering computational work of Charney, Fjörtoft, and Von Neumann (see Charney et al. (1950)), numerical weather prediction (NWP) has proven to present formidable mathematical challenges. In particular, many dynamic models of weather phenomena exhibit multiscale and turbulent features which have been known since the seminal work of Lorenz (1963) to lead to a sensitive dependence on initial conditions. As a consequence, the uncertainties present in a set of initial observations grow exponentially in time under these models, bounding the predictive power of most numerical weather forecasts to medium-range time scales ( 14 days). This chaotic behavior is exacerbated by the computational reality that simulations of the relevant physics can only capture a finite range of scales, so that the physical influence of unresolved scales is either ignored or approximated by subgrid closure models. In recent years, there has been an explosion of interest surrounding data-driven approaches to weather modeling (see, e.g., Rasp et al. (2024) and Karlbauer et al. (2024) for a discussion and recent benchmarks). In contrast to traditional NWP, which relies on numerical simulations of physics-based weather models, these novel data-driven approaches learn effective weather models directly from empirical data.

Limited accessibility to high field MRI scanners (such as 7T, 11T) has motivated the development of post-processing methods to improve low field images. Several existing post-processing methods have shown the feasibility to improve 3T images to produce 7T-like images [3,18]. It has been observed that improving lower field (LF, <=1.5T) images comes with additional challenges due to poor image quality such as the function mapping 1.5T and higher field (HF, 3T) images is more complex than the function relating 3T and 7T images [10]. Except for [10], no method has been addressed to improve <=1.5T MRI images. Further, most of the existing methods [3,18] including [10] require example images, and also often rely on pixel to pixel correspondences between LF and HF images which are usually inaccurate for <=1.5T images. The focus of this paper is to address the unsupervised framework for quality improvement of 1.5T images and avoid the expensive requirements of example images and associated image registration. The LF and HF images are assumed to be related by a linear transformation (LT). The unknown HF image and unknown LT are estimated in alternate minimization framework. Further, a physics based constraint is proposed that provides an additional non-linear function relating LF and HF images in order to achieve the desired high contrast in estimated HF image. The experimental results demonstrate that the proposed approach provides processed 1.5T images, i.e., estimated 3T-like images with improved image quality, and is comparably better than the existing methods addressing similar problems. The improvement in image quality is also shown to provide better tissue segmentation and volume quantification as compared to scanner acquired 1.5T images.

Physics-based dynamic models (PBDMs) are simplified representations of complex dynamical systems. PBDMs take specific processes within a complex system and assign a fragment of variables and an accompanying set of parameters to depict the processes. As this often leads to suboptimal parameterisation of the system, a key challenge requires refining the empirical parameters and variables to reduce uncertainties while maintaining the model's explainability and enhancing its predictive accuracy. We demonstrate that a hybrid mosquito population dynamics model, which integrates a PBDM with Physics-Informed Neural Networks (PINN), retains the explainability of the PBDM by incorporating the PINN-learned model parameters in place of its empirical counterparts. Specifically, we address the limitations of traditional PBDMs by modelling the parameters of larva and pupa development rates using a PINN that encodes complex, learned interactions of air temperature, precipitation and humidity. Our results demonstrate improved mosquito population simulations including the difficult-to-predict mosquito population peaks. This opens the possibility of hybridisation concept application on other complex systems based on PBDMs such as cancer growth to address the challenges posed by scarce and noisy data, and to numerical weather prediction and climate modelling to overcome the gap between physics-based and data-driven weather prediction models. Keywords: hybridisation, physics-based dynamic models, physics-informed neural networks (PINN), hybrid dynamic model, mosquito population modelling 1 Introduction Physics-based dynamic models (PBDMs) are widely used in research and technology, from predicting air temperature to modelling COVID-19 spread and cancer cell development.

In science and engineering, machine learning techniques are increasingly successful in physical systems modeling (predicting future states of physical systems). Effectively integrating PDE loss as a constraint of system transition can improve the model's prediction by overcoming generalization issues due to data scarcity, especially when data acquisition is costly. However, in many real-world scenarios, due to sensor limitations, the data we can obtain is often only partial observation, making the calculation of PDE loss seem to be infeasible, as the PDE loss heavily relies on high-resolution states. We carefully study this problem and propose a novel framework named Re-enable PDE Loss under Partial Observation (RPLPO). The key idea is that although enabling PDE loss to constrain system transition solely is infeasible, we can re-enable PDE loss by reconstructing the learnable high-resolution state and constraining system transition simultaneously. Specifically, RPLPO combines an encoding module for reconstructing learnable high-resolution states with a transition module for predicting future states. The two modules are jointly trained by data and PDE loss. We conduct experiments in various physical systems to demonstrate that RPLPO has significant improvement in generalization, even when observation is sparse, irregular, noisy, and PDE is inaccurate.

The growing threats of uncertainties, anomalies, and cyberattacks on power grids are driving a critical need to advance situational awareness which allows system operators to form a complete and accurate picture of the present and future state. Simulation and estimation are foundational tools in this process. However, existing tools lack the robustness and efficiency required to achieve the level of situational awareness needed for the ever-evolving threat landscape. Industry-standard (steady-state) simulators are not robust to blackouts, often leading to non-converging or non-actionable results. Estimation tools lack robustness to anomalous data, returning erroneous system states. Efficiency is the other major concern as nonlinearities and scalability issues make large systems slow to converge. This thesis addresses robustness and efficiency gaps through a dual-fold contribution. We first address the inherent limitations in the existing physics-based and data-driven worlds; and then transcend the boundaries of conventional algorithmic design in the direction of a new paradigm -- Physics-ML Synergy -- which integrates the strengths of the two worlds. Our approaches are built on circuit formulation which provides a unified framework that applies to both transmission and distribution. Sparse optimization acts as the key enabler to make these tools intrinsically robust and immune to random threats, pinpointing dominant sources of (random) blackouts and data errors. Further, we explore sparsity-exploiting optimizations to develop lightweight ML models whose prediction and detection capabilities are a complement to physics-based tools; and whose lightweight designs advance generalization and scalability. Finally, Physics-ML Synergy brings robustness and efficiency further against targeted cyberthreats, by interconnecting our physics-based tools with lightweight ML.