finite-difference jacobian estimation

This work addresses the reduced robustness of projection-based reduced-order models (ROMs)—such as Proper Orthogonal Decomposition (POD) and Neural Ordinary Differential Equations (Neural ODEs)—in high-dimensional dynamical systems under input data perturbations. We propose a novel training framework that synergistically integrates variational data assimilation (VDA) with supervised learning. Crucially, the VDA mechanism is embedded into the ROM training pipeline to enable dynamic correction of input disturbances, thereby significantly enhancing model stability and prediction accuracy under noisy conditions. The framework is model-agnostic and seamlessly accommodates diverse ROM paradigms, including POD and Neural ODEs. Experimental evaluation on graph-structured dynamical systems demonstrates a 42% reduction in mean prediction error under perturbations. Cross-model generalization tests further confirm consistent and robust performance improvements across different ROM architectures.

This work addresses the challenge of estimating gradients of distribution parameters for random vectors in black-box simulators or expensive physical simulations. We propose Distributional Sensitivity Analysis (DistroSA), the first method to leverage derivatives of the conditional distribution inverse mapping for arbitrary-dimensional random vectors, combined with diagonal Jacobian approximation and four second-order numerical algorithms. DistroSA enables differentiable inference without requiring explicit model knowledge, prior sampling mechanisms, or high-dimensional integration. The resulting differentiable sampling subroutine framework is compatible with automatic differentiation and deep learning platforms, supporting efficient gradient propagation even when closed-form solutions are unavailable. Experiments validate its theoretical correctness and numerical robustness, demonstrating successful application to uncertainty quantification and parameter inversion for nuclear physics quantum correlation functions. The open-source DistroSA package ensures full reproducibility.

研究通过引入敏感性约束的神经算子（SC-NOs），利用求解器导出的雅可比矩阵监督，提高高维偏微分方程系统正向和反向建模的效率与准确性。

Gradient-enhanced global sensitivity analysis (GSA) faces significant challenges under data-scarce conditions, where accurate estimation of Sobol′ indices requires both high-fidelity gradient information and efficient sample utilization. Method: This paper proposes a surrogate modeling framework based on the Poincaré chaos expansion. Unlike conventional orthogonal bases, the Poincaré basis is uniquely characterized by simultaneous orthogonality of both basis functions and their first-order derivatives, derived from a Sturm–Liouville eigenvalue problem and compatible with arbitrary probability measures and weighting schemes. The method integrates a weighted Poincaré inequality, sparse gradient-enhanced regression, and a derivative-driven sensitivity weighting mechanism. Contribution/Results: It enables highly accurate and computationally efficient estimation of Sobol′ indices. Evaluated on a flood modeling case study, the approach achieves reliable sensitivity ranking using only a small number of samples, demonstrating strong adaptability to complex real-world problems and significant computational advantages over existing methods.

This work addresses the challenges of stability degradation and accuracy loss in neural differential equations during long-term integration, as well as the high computational cost associated with training on long trajectories. To overcome these issues, the authors propose two low-cost Jacobian regularization strategies: one that directly computes directional derivatives when the dynamics are known, and another that employs finite-difference approximations when they are not. These approaches significantly enhance the long-term simulation stability of models trained on short trajectories while substantially reducing training overhead. The method demonstrates robustness and scalability across multiple ordinary and partial differential equation systems, offering an efficient pathway for learning large-scale dynamical systems.

This work addresses the inefficiency and numerical instability of existing trajectory optimization methods, which typically rely on numerical or automatic differentiation to compute Jacobians of high-order time derivatives—such as jerk and rate of force change—while neglecting the structural properties of multibody systems. The authors propose a novel analytical framework that explicitly models physical quantities and their higher-order derivatives by leveraging the inherent structure of multibody dynamics. For the first time, they derive structured Jacobian matrices with respect to generalized coordinates and their higher-order time derivatives. By integrating analytical differentiation with multibody dynamics, the method enables efficient and scalable forward and inverse optimization. It significantly improves computational efficiency and numerical stability compared to conventional approaches and demonstrates success in accurately recovering cost function weights from motion data in inverse optimization tasks.

In black-box optimization where only noisy function evaluations are available, finite-difference gradient estimators suffer from unstable performance due to their reliance on perturbation step sizes that depend on unknown model constants. This work proposes a pilot calibration mechanism requiring minimal simulation budget to adaptively estimate these critical parameters and set the perturbation step size accordingly. For the first time, it is theoretically established that this strategy enables finite-difference estimators to achieve the same first-order mean squared error as if the optimal step size were known a priori. The approach is broadly applicable across diverse objective functions and difference schemes, and numerical experiments demonstrate its robustness and near-oracle optimality, significantly outperforming conventional methods that require manual tuning.

This work addresses the lack of efficient, end-to-end differentiable solvers for differential-algebraic equations (DAEs) arising from multiphysics systems, which hinders parameter inversion and optimal control. We present the first natively JAX-based differentiable DAE solver that unifies forward simulation with reverse-mode automatic differentiation. The solver integrates adaptive BDF, Radau, and Rosenbrock methods, combined with Pantelides index reduction and dummy derivative techniques. By freezing the forward time-step grid and re-solving a variable-step BDF-2 scheme on this fixed mesh, gradient propagation becomes highly efficient, enabling DAEs to serve as differentiable primitives. A single call to jax.grad computes full gradients, and wall-clock time remains nearly constant when scaling batched parameter sweeps from 1 to 1,000, dramatically improving differentiable computation efficiency for multiphysics DAE systems.

This work addresses the sensitivity to initial guesses and high computational cost of Newton’s method for solving nonlinear parameterized partial differential equations. The authors propose a two-stage initialization strategy: first, by leveraging parameter sampling and a precomputed solution library, they construct two complementary feature spaces—solution manifold and corrected search directions—from discrete Newton trajectories; second, a regression model predicts a surrogate initial guess, which is then refined via lightweight GMRES-based residual minimization to yield a high-quality starting point. Operating under a weakly intrusive framework, this approach significantly accelerates high-fidelity Newton iterations, markedly reducing both iteration counts and total CPU time on benchmark PDE problems, outperforming existing methods that rely solely on surrogate-based initialization.

This study addresses the high sensitivity to noise inherent in differential-algebraic parameter estimation, which arises from its reliance on exact derivatives and severely limits practical applicability. To overcome this limitation, this work integrates Gaussian process regression (GPR) into the differential-algebraic framework, proposing a robust parameter estimation method for ordinary differential equations that synergizes GPR with algebraic elimination. Furthermore, a first-order error analysis theoretical framework is established to characterize noise propagation and parameter sensitivity. Benchmark evaluations demonstrate that the proposed approach achieves state-of-the-art performance, successfully recovering parameters within a 10% relative error in 88.5% of experimental runs. These results confirm that the method effectively resolves the challenge of parameter identification from noisy observational data.