nullspace projection

For polynomial dynamical systems, this work addresses the core subproblem of differential elimination—computing the minimal-order projection differential equation satisfied by a single coordinate variable. Method: Leveraging tools from differential algebra, Newton polytope theory, and sparse polynomial interpolation, we derive a tight support bound for the Newton polytope of this minimal equation and develop the first scalable evaluation-interpolation algorithm. Contribution/Results: We prove that the bound is optimal in over 50% of cases. Our implementation efficiently solves large-scale instances that existing differential elimination software cannot handle, achieving significant breakthroughs in both computational feasibility and runtime efficiency. The algorithm scales to problems previously deemed intractable, demonstrating superior performance on benchmarks involving high-degree, high-dimensional polynomial systems.

This paper addresses the efficient computation of robust controllable sets (RCS) for discrete-time linear systems subject to additive uncertainties. We propose a novel projection-free and convex-optimization-free algorithm capable of computing inner and outer approximations of the RCS under both ellipsoidal and constrained-zonotopic uncertainty descriptions. Key contributions include: (1) the first closed-form, least-squares-based approximation of the Pontryagin difference; (2) the first fully projection-free RCS computation framework for constrained-zonotopic uncertainties; and (3) guaranteed convergence of all approximations to the exact RCS under verifiable conditions. The method integrates constrained-zonotope representations, convex polyhedral outer approximations, and robust control-invariant set synthesis. Experiments demonstrate scalability: computing a 20-step inner approximation of the RCS for a 100-dimensional system takes under 15 seconds. The approach is successfully applied to safety-critical abort trajectory design for spacecraft performing near-straight-line halo-orbit rendezvous under uncertainty.

This work proposes NP-PIELM, a novel approach to physics-informed extreme learning machines (PIELM) that overcomes the limitations of traditional penalty-based formulations, which suffer from limited accuracy and high sensitivity to hyperparameters due to approximate enforcement of boundary and initial conditions. By introducing a nullspace projection in the coefficient space, NP-PIELM reformulates the constrained optimization problem as an unconstrained least-squares problem, enabling exact satisfaction of boundary conditions at discrete collocation points. This method achieves strict enforcement of boundary constraints within the PIELM framework for the first time—without requiring penalty parameters, dual variables, or problem-specific constructions—while preserving the single-shot training efficiency inherent to ELMs. Numerical experiments demonstrate that NP-PIELM delivers high accuracy and robustness across elliptic and parabolic partial differential equations, complex geometries, and mixed boundary conditions.

This work addresses the challenge of parameter optimization in computer model calibration, where the goal is to minimize discrepancies between multidimensional model outputs and observed data. The authors propose a novel root-finding paradigm that reformulates calibration as a root-search problem by constructing signed-residual-based Kriging or stochastic Kriging surrogates. Their approach integrates a sequential search space reduction strategy with a new acquisition function compatible with first-order optimizers. Notably, it guarantees algorithmic robustness even when roots may not exist—a scenario unaddressed by prior methods. Empirical evaluations demonstrate that the proposed method significantly outperforms conventional calibration techniques across both data-driven and physics-based modeling tasks, achieving higher computational efficiency while enhancing solution robustness.

In robot state estimation, anisotropic (matrix-weighted) noise causes conventional convex relaxations to lose tightness, thereby invalidating certification guarantees. Method: This paper identifies the fundamental mechanism by which matrix weighting undermines the tightness of semidefinite relaxation (SDR), establishes a theoretical link between posterior uncertainty and the certificate matrix, and proposes a novel paradigm—incorporating geometric redundancy constraints—to restore tightness. It further constructs the first tightable semidefinite programming (SDP) formulation for matrix-weighted SLAM. Contribution/Results: We theoretically prove that the proposed SDP admits a tightness bound under low-noise conditions. Extensive simulations and real-world experiments demonstrate that redundancy constraints significantly improve tightness rates; moreover, the new formulation maintains tightness even under extremely low noise, outperforming scalar-weighted approaches in both certification reliability and estimation accuracy.

This work proposes a novel regression framework based on the m-product of tensors to address the curse of dimensionality and overfitting arising from data scarcity in high-dimensional nonlinear regression. By integrating kernel methods with tensor algebra, the approach constructs an efficient regularization mechanism through structured variable separation, circumventing conventional fixed-point iterations while preserving favorable properties of matrix computations. Experimental results on standard benchmarks and dynamical system tasks demonstrate that the method robustly and efficiently handles regression with hundreds of parameters using only small sample sizes, exhibiting strong scalability and practical utility for real-world engineering applications.

This work addresses the challenge of high errors in neural surrogate models for stiff differential-algebraic equations (DAEs), which arise due to algebraic residuals being amplified by stiffness or reliance on costly numerical integration. The authors propose an extended Newton implicit layer that jointly enforces algebraic consistency and quasi-steady-state dimensionality reduction within a single differentiable solve. By predicting only the slow-varying states, the method accurately recovers both fast dynamics and algebraic variables while reducing output dimensionality to the slow subspace. This is the first approach to integrate physics-guided dimensionality reduction with implicit differential operator learning without simulation-based training. Leveraging the implicit function theorem, it derives gradients accounting for stiff coupling, supports compositional modeling of multi-component cascaded systems, and offers theoretical convergence guarantees. Evaluated on a 21-state power inverter DAE, it achieves a mere 1.42% error—significantly outperforming penalty methods (39.3%)—and composes two models into a 44-state system without retraining, yielding 0.72–1.16% error, zero algebraic residual, and 90% in-distribution coverage via conformal prediction.

This work addresses the challenge of globally characterizing the geometric structure of solution manifolds in redundant robotic tasks, which exhibit non-uniqueness and form continuous manifolds in configuration space. Existing approaches struggle to capture these structures comprehensively. The paper proposes a representation-centric implicit modeling paradigm that constructs a scalar field over the configuration space, whose zero-level set precisely coincides with the task-induced solution manifold. By integrating Jacobian-guided neighborhood sampling with implicit neural representations, the method learns a signed distance field of the solution manifold, enabling globally consistent and continuous modeling under arbitrary task mappings—a capability demonstrated for the first time. Experiments on a planar three-link robot and a seven-degree-of-freedom Franka manipulator validate the approach’s ability to accurately reconstruct solution manifolds and generalize across varying task parameters.

This work addresses the long-standing issue of inconsistency in generalized matrix inverses under nonsingular diagonal transformations—a problem particularly critical in robotics, tracking, and control systems where physical unit sensitivity is paramount. The paper introduces a novel generalized inverse that, for the first time, achieves invariance under arbitrary nonsingular diagonal transformations, thereby rigorously preserving the physical units of state-space variables. This new inverse, together with the Moore–Penrose and Drazin inverses, forms a complete triad encompassing the principal linear system transformations. The framework is further extended to unit-consistent matrix factorizations. Grounded in matrix analysis theory, the authors develop a new algebraic construction method, successfully applied across multiple engineering domains, providing a robust mathematical foundation for unit-preserving modeling and significantly advancing the theoretical completeness of generalized inverses.

This study addresses the computational challenge of efficiently simulating high-dimensional, nonlinear aeroelastic–flight dynamics coupled systems. The authors propose a general nonlinear model order reduction framework that constructs a second-order Taylor expansion of the residual around equilibrium points—sufficient to accurately represent cubic nonlinearities without requiring third-order terms—and employs a bi-orthogonal low-dimensional subspace spanned by the left and right eigenvectors of the Jacobian matrix to achieve an optimal projection. By integrating a matrix-free finite difference approximation, the method avoids dependence on the full-order model’s internal structure. Validated across three test cases of increasing complexity, the approach reduces system dimensionality from thousands to single digits, achieving speedups of up to 600× while accurately capturing strong nonlinear dynamic behaviors such as large deformations exceeding 10% of the wingspan.