stabilized variable transformation

This work proposes a stabilization framework based on score-based generative models to address non-physical instabilities and structural distortions commonly encountered in the numerical solution of time-dependent partial differential equations (PDEs). For the first time, score-based generative models are introduced into PDE numerical stabilization, where a conditional stabilizing operator with manifold-contracting properties is constructed by learning the physically admissible solution manifold. This operator corrects intermediate solutions during time integration. Numerical experiments demonstrate that the method significantly enhances robustness for convection, Korteweg–de Vries (KdV), nonlinear Schrödinger, and Burgers equations, effectively suppressing spurious oscillations while preserving essential dynamical features.

This work proposes a systematic framework to enhance the computational efficiency and numerical stability of evaluating high-degree matrix polynomials. Specifically, for polynomial degrees eight and higher, the method generates and validates stable coefficient sets that reduce the number of required matrix multiplications by one compared to the classical Paterson–Stockmeyer scheme. To address instability issues in the original formulation, the authors introduce structural variants and design a reliability metric to assess the expected numerical accuracy of candidate coefficient sets. Nonlinear polynomial systems are solved using variable-precision arithmetic (VPA), and an in-house tool, MatrixPolEval1, enables efficient screening and validation. Applied to matrix exponentials and geometric series, the approach achieves a saving of one matrix multiplication while maintaining comparable numerical accuracy.

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.

Stability selection lacks a principled framework for assessing the global stability of selected variables. Method: We propose a global robustness measure based on a stability estimator—first applied to global stability evaluation and adaptive regularization parameter selection. We derive its asymptotic distribution and theoretically determine the minimum required subsample size. Furthermore, we unify the calibration of the decision threshold, expected number of false positives, and regularization strength within a Pareto-optimality framework. Contribution/Results: Our work fills a critical theoretical gap regarding subsample size guidance; enables statistically guaranteed parameter calibration, interpretable quantification of stability, and integrable visualization of results. To facilitate practical implementation, we release an open-source R package, *stabplot*, supporting end-to-end stability analysis.

This work addresses the challenge of convergence failure in inverse parallel solvers for nonlinear systems of equations, which often arises due to oscillatory or chaotic dynamics. To enhance stability, the authors propose an adaptive stabilization mechanism based on the local maximum Lyapunov exponent (LLE). By estimating the LLE via k-nearest neighbors and integrating it with sliding-window micro-time-series analysis, the method enables real-time detection of unstable phases along the solution trajectory. A Lyapunov-guided parameter control strategy is then developed to dynamically adjust solver parameters, thereby reinforcing numerical stability. Experimental results demonstrate strong agreement between theoretical stability diagrams and empirical Lyapunov profiles, confirming that the proposed approach significantly improves the robustness and convergence performance of solvers under perturbed initial conditions.

This work addresses the challenge of selecting the regularization parameter (nugget) in ill-posed linear systems arising in machine learning, where existing adaptive methods lack compatibility with automatic differentiation and suffer from computational inefficiency. To overcome these limitations, we introduce autonugget, a lightweight Python package fully compatible with JAX’s automatic differentiation framework. Our approach uniquely integrates Richardson extrapolation with Tikhonov regularized solutions computed across multiple nugget values, thereby preserving end-to-end differentiability while avoiding the information loss inherent in single-solution strategies. Experimental results demonstrate that autonugget significantly enhances solution accuracy and training stability without compromising rapid prototyping capabilities.

Existing probabilistic ODE solvers struggle to simultaneously achieve numerical stability and scalability when applied to stiff, high-dimensional problems. This work proposes two complementary strategies to address this challenge: first, a matrix-free update mechanism leveraging Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to attain linear computational complexity while ensuring numerical stability; second, an iterative re-linearization scheme that reformulates the solver into a fully implicit form, further enhancing stability without compromising scalability. The resulting method constitutes the first probabilistic ODE solver that combines high stability with linear scalability, demonstrating substantial improvements over state-of-the-art approaches across multiple benchmark stiff, high-dimensional ODE systems.

This work addresses the unclear stability mechanisms of zeroth-order (ZO) optimization methods in deep learning, particularly the lack of theoretical characterization regarding the relationship between step size and the Hessian spectrum. Through mean-square linear stability analysis, we reveal for the first time that the stability condition of ZO methods depends on the full Hessian spectrum rather than solely on its largest eigenvalue—as is typical for first-order methods. We derive a computable stability boundary requiring only the largest eigenvalue and the trace of the Hessian, and further uncover that large step sizes implicitly regularize the Hessian trace in ZO optimization. These theoretical findings apply to ZO-GD, ZO-GDM, and ZO-Adam, and are empirically validated across diverse deep learning tasks, where these methods operate near the predicted stability edge.

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 unclear relationship between continuous theory and discrete implementation in neural operators for solving partial differential equations, particularly concerning stability and discretization error. For the first time, rigorous discretization error bounds are established for State-Space Neural Operators (SS-NOs) and Fourier Neural Operators (FNOs). By leveraging functional analysis, the regularity of solutions is explicitly linked to input discretization, and Input-to-State Stability (ISS) theory is introduced to quantify how discretization affects stability in the continuous domain. Numerical experiments on one- and two-dimensional benchmark problems validate the tightness of the derived theoretical bounds, demonstrating that SS-NOs exhibit both robustness and numerical stability across varying resolutions.