We introduce Neural Green's Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green's functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green's Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green's Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green's Function outperforms state-of-the-art neural operators, achieving an average error reduction of 13.9\% across five shape categories, while being up to 350 times faster than a numerical solver that requires computationally expensive meshing.

Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm parameters using problem instances drawn from the problem distribution of the target application domain. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piece-wise and discontinuity structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the Pfaffian GJ framework, an extension of the classical GJ framework, capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a refined piece-wise structure, which automatically translates to learning guarantees using our proposed framework. For the online learning setting, we provide a new tool for verifying dispersion property of a sequence of loss functions. This sufficient condition allows no-regret learning for sequences of piece-wise structured loss functions where the piece-wise structure involves Pfaffian transition boundaries.

Neural operators such as the Fourier Neural Operator (FNO) have been shown to provide resolution-independent deep learning models that can learn mappings between function spaces. For example, an initial condition can be mapped to the solution of a partial differential equation (PDE) at a future time-step using a neural operator. Despite the popularity of neural operators, their use to predict solution functions over a domain given only data over the boundary (such as a spatially varying Dirichlet boundary condition) remains unexplored. In this paper, we refer to such problems as boundary-to-domain problems; they have a wide range of applications in areas such as fluid mechanics, solid mechanics, heat transfer etc. We present a novel FNO-based architecture, named Lifting Product FNO (or LP-FNO) which can map arbitrary boundary functions defined on the lower-dimensional boundary to a solution in the entire domain. Specifically, two FNOs defined on the lower-dimensional boundary are lifted into the higher dimensional domain using our proposed lifting product layer. We demonstrate the efficacy and resolution independence of the proposed LP-FNO for the 2D Poisson equation.

Passivity is necessary for robots to fluidly collaborate and interact with humans physically. Nevertheless, due to the unconstrained nature of passivity-based impedance control laws, the robot is vulnerable to infeasible and unsafe configurations upon physical perturbations. In this paper, we propose a novel control architecture that allows a torque-controlled robot to guarantee safety constraints such as kinematic limits, self-collisions, external collisions and singularities and is passive only when feasible. This is achieved by constraining a dynamical system based impedance control law with a relaxed hierarchical control barrier function quadratic program subject to multiple concurrent, possibly contradicting, constraints. Joint space constraints are formulated from efficient data-driven self- and external C^2 collision boundary functions. We theoretically prove constraint satisfaction and show that the robot is passive when feasible. Our approach is validated in simulation and real robot experiments on a 7DoF Franka Research 3 manipulator.

A realistic human kinematic model that satisfies anatomical constraints is essential for human-robot interaction, biomechanics and robot-assisted rehabilitation. Modeling realistic joint constraints, however, is challenging as human arm motion is constrained by joint limits, inter- and intra-joint dependencies, self-collisions, individual capabilities and muscular or neurological constraints which are difficult to represent. Hence, physicians and researchers have relied on simple box-constraints, ignoring important anatomical factors. In this paper, we propose a data-driven method to learn realistic anatomically constrained upper-limb range of motion (RoM) boundaries from motion capture data. This is achieved by fitting a one-class support vector machine to a dataset of upper-limb joint space exploration motions with an efficient hyper-parameter tuning scheme. Our approach outperforms similar works focused on valid RoM learning. Further, we propose an impairment index (II) metric that offers a quantitative assessment of capability/impairment when comparing healthy and impaired arms. We validate the metric on healthy subjects physically constrained to emulate hemiplegia and different disability levels as stroke patients.

We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in $n$. We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU networks with quadratic depth and linear width in the space dimension. We also investigate the approximation error of one of these functions by shallower networks and prove a separation result. The main difference between our functions and other constructions is their practical interest: they arise in the scope of channel coding. Hence, computing such functions amounts to performing a decoding operation.

Our work deals with the important problem of globally characterizing truthful mechanisms where players have multi-parameter, additive valuations, like scheduling unrelated machines or additive combinatorial auctions. Very few mechanisms are known for these settings and the question is: Can we prove that no other truthful mechanisms exist? We characterize truthful mechanisms for n players and 2 tasks or items, as either task-independent, or a player-grouping minimizer, a new class of mechanisms we discover, which generalizes affine minimizers. We assume decisiveness, strong monotonicity and that the truthful payments (The (normalized) payments are uniquely determined by the allocation function of the mechanism; thus the assumptions concern properties of the allocation.) are continuous functions of players' bids.