Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the solution operators of partial differential equations (PDEs). These methods can also be used to develop black-box simulators to model system behavior from experimental data, even without a known mathematical model. In this article, we begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field. We also discuss PDE-specific operator learning, outlining strategies for incorporating physical and mathematical constraints into architecture design and training processes. Finally, we end by highlighting key future directions such as active data collection and the development of rigorous uncertainty quantification frameworks.

Algorithms with predictions are gaining traction across various domains, as a way to surpass traditional worst-case bounds through (machine-learned) advice. We study the canonical problem of k-facility location mechanism design, where the n agents are strategic and might misreport their locations. We receive a prediction for each agent's location, and these predictions are crucially allowed to be only "mostly" and "approximately" correct (M

Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling problems on rectangular domains. To lift such a restriction and permit FFT on irregular geometries as well as topology changes, we introduce domain agnostic Fourier neural operator (DAFNO), a novel neural operator architecture for learning surrogates with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of FNOs, and leverage FFT to achieve rapid computations, in such a way that the geometric information is explicitly encoded in the architecture. In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as compared to baseline neural operator models on two benchmark datasets of material modeling and airfoil simulation. To further demonstrate the capability and generalizability of DAFNO in handling complex domains with topology changes, we consider a brittle material fracture evolution problem. With only one training crack simulation sample, DAFNO has achieved generalizability to unseen loading scenarios and substantially different crack patterns from the trained scenario.

We consider a dynamic mechanism design problem where an auctioneer sells an indivisible good to two groups of buyers in every round, for a total of T rounds. The auctioneer aims to maximize their discounted overall revenue while adhering to a fairness constraint that guarantees a minimum average allocation for each group. We begin by studying the static case (T = 1) and establish that the optimal mechanism involves two types of subsidization: one that increases the overall probability of allocation to all buyers, and another that favors the group which otherwise has a lower probability of winning the item. We then extend our results to the dynamic case by characterizing a set of recursive functions that determine the optimal allocation and payments in each round. Notably, our results establish that in the dynamic case, the seller, on the one hand, commits to a participation reward to incentivize truth-telling, and, on the other hand, charges an entry fee for every round. Moreover, the optimal allocation once more involves subsidization in favor of one group, where the extent of subsidization depends on the difference in future utilities for both the seller and buyers when allocating the item to one group versus the other. Finally, we present an approximation scheme to solve the recursive equations and determine an approximately optimal and fair allocation efficiently.

We develop a versatile new methodology for multidimensional mechanism design that incorporates side information about agent types to generate high social welfare and high revenue simultaneously. Prominent sources of side information in practice include predictions from a machine-learning model trained on historical agent data, advice from domain experts, and even the mechanism designer's own gut instinct. In this paper we adopt a prior-free perspective that makes no assumptions on the correctness, accuracy, or source of the side information.

Spatio-temporal (ST) prediction has garnered a De facto attention in earth sciences, such as meteorological prediction, human mobility perception. However, the scarcity of data coupled with the high expenses involved in sensor deployment results in notable data imbalances. Furthermore, models that are excessively customized and devoid of causal connections further undermine the generalizability and interpretability. To this end, we establish a framework for ST predictions from a causal perspective, termed CaPaint, which targets to identify causal regions in data and endow model with causal reasoning ability in a two-stage process. Going beyond this process, we build on the front door adjustment as the theoretical foundation to specifically address the sub-regions identified as non-causal in the upstream phase.

S automation advances, robots are increasingly utilized for complex tasks, reducing manual labor in hazardous environments while improving efficiency, precision, and cost-effectiveness [1]. However, real-world robotic applications require seamless interaction with deformable objects, which presents significant challenges due to material flexibility and unpredictable shape changes [2]. Unlike rigid object manipulation, deformable object manipulation (DOM) requires real-time adaptive control to compensate for continuous state variations and external forces. Traditional physics-based control models, such as mass-spring systems and finite element methods [3], [4], [5], attempt to model deformable object behavior but often fall short in real-world applications due to the sensitvity of control parameters and the difficulty of modeling complex contact dynamics. To address these limitations, recent research has shifted toward machine learning and data-driven approaches, where robots learn from sensor feedback or demonstrations rather than relying on hard-coded models [6]. Predictive learning models [7], [8], [9] have proven effective for latent space learning and object behavior forecasting, improving adaptability across applications such as fabric repositioning [10], crop harvesting [11], [12], medical robotics [13], and deformable linear object manipulation [14], [15]. While significant progress has been made in DOM, little research has focused on deformable tool manipulation (DTM), which introduces additional complexities such as bending dynamics, force regulation, and stability issues.

The ability to perform causal reasoning is widely considered a core feature of intelligence. In this work, we investigate whether large language models (LLMs) can coherently reason about causality. Much of the existing work in natural language processing (NLP) focuses on evaluating commonsense causal reasoning in LLMs, thus failing to assess whether a model can perform causal inference in accordance with a set of well-defined formal rules. To address this, we propose a new NLP task, causal inference in natural language, inspired by the "causal inference engine" postulated by Judea Pearl et al.

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.