To address this, we leverage parametric modular structure to learn component-level surrogates, enabling cheaper high-fidelity simulation. We use a neural network to model the stored potential energy in a component given boundary conditions.

Abstract-- Hard-constraint trajectory planners often rely on commercial solvers and demand substantial computational resources. Existing soft-constraint methods achieve faster computation, but either (1) decouple spatial and temporal optimization or (2) restrict the search space. T o overcome these limitations, we introduce MIGHTY, a Hermite spline-based planner that performs spatiotemporal optimization while fully leveraging the continuous search space of a spline. In simulation, MIGHTY achieves a 9.3% reduction in computation time and a 13.1% reduction in travel time over state-of-the-art baselines, with a 100% success rate. In hardware, MIGHTY completes multiple high-speed flights up to 6.7 m/s in a cluttered static environment and long-duration flights with dynamically added obstacles. Trajectory planning for autonomous navigation has been extensively studied, with a wide variety of parameterizations and formulations [3], [6], [7], [9], [12], [14]-[17], [19]-[23].

Additionally, we introduce a Bézier-boundary gradient approximation strategy within our framework to keep the "differentiability"

Data-driven discovery of governing equations from data remains a fundamental challenge in nonlinear dynamics. Although sparse regression techniques have advanced system identification, they struggle with rational functions and noise sensitivity in complex mechanical systems. The Lagrangian formalism offers a promising alternative, as it typically avoids rational expressions and provides a more concise representation of system dynamics. However, existing Lagrangian identification methods are significantly affected by measurement noise and limited data availability. This paper presents a novel differentiable sparse identification framework that addresses these limitations through three key contributions: (1) the first integration of cubic B-Spline approximation into Lagrangian system identification, enabling accurate representation of complex nonlinearities, (2) a robust equation discovery mechanism that effectively utilizes measurements while incorporating known physical constraints, (3) a recursive derivative computation scheme based on B-spline basis functions, effectively constraining higher-order derivatives and reducing noise sensitivity on second-order dynamical systems. The proposed method demonstrates superior performance and enables more accurate and reliable extraction of physical laws from noisy data, particularly in complex mechanical systems compared to baseline methods.

In order to provide more information and explanation about our training data and evaluation benchmark, we introduce our synthetic samples with respect to the length and density of point trajectory.

First, the point trajectories generated by these approaches only reflect the possible locations at certain moments instead of real motion trends because the frame-to-frame correspondence is temporally discrete and lacks enough inter-frame information to describe continuous point motion.

Modern approximations to Gaussian processes are suitable for "tall data", with a cost that scales well in the number of observations, but under-performs on "wide

The proof follows similar arguments as in Proposition 4 from Blondel et al. [ 2020 ]. We now derive differentiable forms of generalized piecewise d -polynomial regression, which is used in applications such as spline fittings. Our 1D piecewise spline approximation can be (heuristically) extended to 2D data. We consider the problem of image segmentation, which can be viewed as representing the domain of an image into a disjoint union of subsets. Instead, we leverage connected-component algorithms (such as Hoshen-Kopelman, or other, techniques [ Wu et al., 2005 ]) to produce a partition, and the predicted output is a piecewise constant image with C.2 NURBS derivatives We rewrite the NURBS formulation as follows: S ( u, v)= NR ( u, v) w ( u, v) (20) where, NR ( u, v)= For simplicity, we will stick to 1D NURBS curves.

Differentiable programming has been a paradigm shift in algorithm design.

In this paper, we present DiffSketcher, an innovative algorithm that creates vectorized free-hand sketches using natural language input.