Scientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem at hand, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularized. Such a diffusion has been triggered by a huge availability of data (the so-called big data), an increasingly cheap computing power, and the development of powerful machine learning algorithms. SciML leverages the physical awareness of physics-based models and, at the same time, the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into machine learning algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and non-linear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and machine learning algorithms, and presenting the most popular machine learning architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by partial differential equations. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socio-economic importance that poses numerous challenges on both the mathematical and computational fronts. The corresponding mathematical model is a complex system of non-linear ordinary and partial differential equations describing the electromechanics, valve dynamics, blood circulation, perfusion in the coronary tree, and torso potential. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models.

Healthcare decision-making requires not only accurate predictions but also insights into how factors influence patient outcomes. While traditional Machine Learning (ML) models excel at predicting outcomes, such as identifying high risk patients, they are limited in addressing what-if questions about interventions. This study introduces the Probabilistic Causal Fusion (PCF) framework, which integrates Causal Bayesian Networks (CBNs) and Probability Trees (PTrees) to extend beyond predictions. PCF leverages causal relationships from CBNs to structure PTrees, enabling both the quantification of factor impacts and simulation of hypothetical interventions. PCF was validated on three real-world healthcare datasets i.e. MIMIC-IV, Framingham Heart Study, and Diabetes, chosen for their clinically diverse variables. It demonstrated predictive performance comparable to traditional ML models while providing additional causal reasoning capabilities. To enhance interpretability, PCF incorporates sensitivity analysis and SHapley Additive exPlanations (SHAP). Sensitivity analysis quantifies the influence of causal parameters on outcomes such as Length of Stay (LOS), Coronary Heart Disease (CHD), and Diabetes, while SHAP highlights the importance of individual features in predictive modeling. By combining causal reasoning with predictive modeling, PCF bridges the gap between clinical intuition and data-driven insights. Its ability to uncover relationships between modifiable factors and simulate hypothetical scenarios provides clinicians with a clearer understanding of causal pathways. This approach supports more informed, evidence-based decision-making, offering a robust framework for addressing complex questions in diverse healthcare settings.

Recent advancements in machine learning-based methods have demonstrated great potential for improved property prediction in material science. However, reliable estimation of the confidence intervals for the predicted values remains a challenge, due to the inherent complexities in material modeling. This study introduces a novel approach for uncertainty quantification in fatigue life prediction of metal materials based on integrating knowledge from physics-based fatigue life models and machine learning models. The proposed approach employs physics-based input features estimated using the Basquin fatigue model to augment the experimentally collected data of fatigue life. Furthermore, a physics-informed loss function that enforces boundary constraints for the estimated fatigue life of considered materials is introduced for the neural network models. Experimental validation on datasets comprising collected data from fatigue life tests for Titanium alloys and Carbon steel alloys demonstrates the effectiveness of the proposed approach. The synergy between physics-based models and data-driven models enhances the consistency in predicted values and improves uncertainty interval estimates.

This submission drew a great deal of discussion -- primarily on the point of the role of learning. All reviewers agreed that the approach had the potential to learn interesting, non-trivial things but did not feel the the current experiments demonstrated these effectively -- despite strong performance on the task. Some examples of questions that were not answered by the main draft but came up in the discussion: [Training Data] The training data provides edges in the dependency graph, subgoals, and predicate value -- image pairs. One question was whether the union of the seen dependency graph constituted the entire true underlying graph. Similarly, do all predicate-object pairs occur?

The maritime industry aims towards a sustainable future, which requires significant improvements in operational efficiency. Current approaches focus on minimising fuel consumption and emissions through greater autonomy. Efficient and safe autonomous navigation requires high-fidelity ship motion models applicable to real-world conditions. Although physics-based ship motion models can predict ships' motion with sub-second resolution, their validation in real-world conditions is rarely found in the literature. This study presents a physics-based 3D dynamics motion model that is tailored to a container-ship, and compares its predictions against real-world voyages. The model integrates vessel motion over time and accounts for its hydrodynamic behavior under different environmental conditions. The model's predictions are evaluated against real vessel data both visually and using multiple distance measures. Both methodologies demonstrate that the model's predictions align closely with the real-world trajectories of the container-ship.

This paper deals with the sample complexity of automated mechanism design for the problem of maximizing the revenue in a combinatorial auction (CA). Given a class of auction mechanisms, the automated mechanism design takes as input samples from the bidders' valuation distribution (which, in practice, may be the history records from the previous auctions), and output the choice of auction mechanism with high revenue. This work presents several upper bounds on the sample complexities for various auction classes. Although Morgenstern and Roughgarden (reference [19] in this paper) studied the same problem of bounding the sample complexities of CA, their work only deals with "simple auctions" which can be reduced to the single-bidder setting. In contrast, this paper studies the hierarchy of deterministic CA families consists of VCG-based mechanisms.

Pre-trained machine learning (ML) models have shown great performance for awide range of applications, in particular in natural language processing (NLP)and computer vision (CV). Here, we study how pre-training could be used forscientific machine learning (SciML) applications, specifically in the context oftransfer learning. We study the transfer behavior of these models as (i) the pretrainedmodel size is scaled, (ii) the downstream training dataset size is scaled,(iii) the physics parameters are systematically pushed out of distribution, and (iv)how a single model pre-trained on a mixture of different physics problems canbe adapted to various downstream applications. We also find that fine-tuning these modelsyields more performance gains as model size increases, compared to training fromscratch on new downstream tasks. These results hold for a broad range of PDElearning tasks.

Generating animation of physics-based characters with intuitive control has long been a desirable task with numerous applications. However, generating physically simulated animations that reflect high-level human instructions remains a difficult problem due to the complexity of physical environments and the richness of human language. In this paper, we present \textbf{InsActor}, a principled generative framework that leverages recent advancements in diffusion-based human motion models to produce instruction-driven animations of physics-based characters.Our framework empowers InsActor to capture complex relationships between high-level human instructions and character motions by employing diffusion policies for flexibly conditioned motion planning.To overcome invalid states and infeasible state transitions in planned motions, InsActor discovers low-level skills and maps plans to latent skill sequences in a compact latent space. Extensive experiments demonstrate that InsActor achieves state-of-the-art results on various tasks, including instruction-driven motion generation and instruction-driven waypoint heading. Notably, the ability of InsActor to generate physically simulated animations using high-level human instructions makes it a valuable tool, particularly in executing long-horizon tasks with a rich set of instructions.

We study collaborative normal mean estimation, where m strategic agents collect i.i.d samples from a normal distribution \mathcal{N}(\mu, \sigma 2) at a cost. They all wish to estimate the mean \mu . By sharing data with each other, agents can obtain better estimates while keeping the cost of data collection small. To facilitate this collaboration, we wish to design mechanisms that encourage agents to collect a sufficient amount of data and share it truthfully, so that they are all better off than working alone. In naive mechanisms, such as simply pooling and sharing all the data, an individual agent might find it beneficial to under-collect and/or fabricate data, which can lead to poor social outcomes.

We study Bayesian automated mechanism design in unstructured dynamic environments, where a principal repeatedly interacts with an agent, and takes actions based on the strategic agent's report of the current state of the world. Both the principal and the agent can have arbitrary and potentially different valuations for the actions taken, possibly also depending on the actual state of the world. Moreover, at any time, the state of the world may evolve arbitrarily depending on the action taken by the principal. The goal is to compute an optimal mechanism which maximizes the principal's utility in the face of the self-interested strategic agent.We give an efficient algorithm for computing optimal mechanisms, with or without payments, under different individual-rationality constraints, when the time horizon is constant. Our algorithm is based on a sophisticated linear program formulation, which can be customized in various ways to accommodate richer constraints.