Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex systems remain unknown due to intricate underlying mechanisms. Recent advancements in machine learning and data science offer a new paradigm for modeling unknown equations from measurement or simulation data. This paradigm shift, known as data-driven discovery or modeling, stands at the forefront of AI for science, with significant progress made in recent years. In this paper, we introduce a systematic framework for data-driven modeling of unknown equations using deep learning. This versatile framework is capable of learning unknown ODEs, PDEs, DAEs, IDEs, SDEs, reduced or partially observed systems, and non-autonomous differential equations. Based on this framework, we have developed Deep Unknown Equations (DUE), an open-source software package designed to facilitate the data-driven modeling of unknown equations using modern deep learning techniques. DUE serves as an educational tool for classroom instruction, enabling students and newcomers to gain hands-on experience with differential equations, data-driven modeling, and contemporary deep learning approaches such as FNN, ResNet, generalized ResNet, operator semigroup networks (OSG-Net), and Transformers. Additionally, DUE is a versatile and accessible toolkit for researchers across various scientific and engineering fields. It is applicable not only for learning unknown equations from data but also for surrogate modeling of known, yet complex, equations that are costly to solve using traditional numerical methods. We provide detailed descriptions of DUE and demonstrate its capabilities through diverse examples, which serve as templates that can be easily adapted for other applications.

Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism$\unicode{x2013}$a powerful tool originally designed for natural language processing$\unicode{x2013}$have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.

Coalition is an important mean of multi-robot systems to collaborate on common tasks. An adaptive coalition strategy is essential for the online performance in dynamic and unknown environments. In this work, the problem of territory defense by large-scale heterogeneous robotic teams is considered. The tasks include exploration, capture of dynamic targets, and perimeter defense over valuable resources. Since each robot can choose among many tasks, it remains a challenging problem to coordinate jointly these robots such that the overall utility is maximized. This work proposes a generic coalition strategy called K-serial stable coalition algorithm. Different from centralized approaches, it is distributed and complete, meaning that only local communication is required and a K-serial Stable solution is ensured. Furthermore, to accelerate adaptation to dynamic targets and resource distribution that are only perceived online, a heterogeneous graph attention network based heuristic is learned to select more appropriate parameters and promising initial solutions during local optimization. Compared with manual heuristics or end-to-end predictors, it is shown to both improve online adaptability and retain the quality guarantee. The proposed methods are validated via large-scale simulations with 170 robots and hardware experiments of 13 robots, against several strong baselines such as GreedyNE and FastMaxSum.

This paper proposes a novel deep learning approach for learning operators in semigroup, with applications to modeling unknown autonomous dynamical systems using time series data collected at varied time lags. It is a sequel to the previous flow map learning (FML) works [T. Qin, K. Wu, and D. Xiu, J. Comput. Phys., 395:620--635, 2019], [K. Wu and D. Xiu, J. Comput. Phys., 408:109307, 2020], and [Z. Chen, V. Churchill, K. Wu, and D. Xiu, J. Comput. Phys., 449:110782, 2022], which focused on learning single evolution operator with a fixed time step. This paper aims to learn a family of evolution operators with variable time steps, which constitute a semigroup for an autonomous system. The semigroup property is very crucial and links the system's evolutionary behaviors across varying time scales, but it was not considered in the previous works. We propose for the first time a framework of embedding the semigroup property into the data-driven learning process, through a novel neural network architecture and new loss functions. The framework is very feasible, can be combined with any suitable neural networks, and is applicable to learning general autonomous ODEs and PDEs. We present the rigorous error estimates and variance analysis to understand the prediction accuracy and robustness of our approach, showing the remarkable advantages of semigroup awareness in our model. Moreover, our approach allows one to arbitrarily choose the time steps for prediction and ensures that the predicted results are well self-matched and consistent. Extensive numerical experiments demonstrate that embedding the semigroup property notably reduces the data dependency of deep learning models and greatly improves the accuracy, robustness, and stability for long-time prediction.

Multi-agent systems can be extremely efficient when working concurrently and collaboratively, e.g., for transportation, maintenance, search and rescue. Coordination of such teams often involves two aspects: (i) selecting appropriate sub-teams for different tasks; (ii) designing collaborative control strategies to execute these tasks. The former aspect can be combinatorial w.r.t. the team size, while the latter requires optimization over joint state-spaces under geometric and dynamic constraints. Existing work often tackles one aspect by assuming the other is given, while ignoring their close dependency. This work formulates such problems as combinatorial-hybrid optimizations (CHO), where both the discrete modes of collaboration and the continuous control parameters are optimized simultaneously and iteratively. The proposed framework consists of two interleaved layers: the dynamic formation of task coalitions and the hybrid optimization of collaborative behaviors. Overall feasibility and costs of different coalitions performing various tasks are approximated at different granularities to improve the computational efficiency. At last, a Nash-stable strategy for both task assignment and execution is derived with provable guarantee on the feasibility and quality. Two non-trivial applications of collaborative transportation and dynamic capture are studied against several baselines.