Stiff systems of ordinary differential equations (ODEs) and sparse training
data are common in scientific problems. This paper describes efficient,
implicit, vectorized methods for integrating stiff systems of ordinary
differential equations through time and calculating parameter gradients with
the adjoint method. The main innovation is to vectorize the problem both over
the number of independent times series and over a batch or "chunk" of
sequential time steps, effectively vectorizing the assembly of the implicit
system of ODEs. The block-bidiagonal structure of the linearized implicit
system for the backward Euler method allows for further vectorization using
parallel cyclic reduction (PCR). Vectorizing over both axes of the input data
provides a higher bandwidth of calculations to the computing device, allowing
even problems with comparatively sparse data to fully utilize modern GPUs and
achieving speed ups of greater than 100x, compared to standard, sequential time
integration. We demonstrate the advantages of implicit, vectorized time
integration with several example problems, drawn from both analytical stiff and
non-stiff ODE models as well as neural ODE models. We also describe and provide
a freely available open-source implementation of the methods developed here.

将刚性系统的常微分方程通过时间的积分与伴随方法计算参数梯度的高效、隐式、矢量化方法在本文中进行了描述。创新之处在于在独立时间序列的数量和连续时间步骤的批次或 “块” 上对问题进行矢量化，从而有效地进行隐式常微分方程系统的组装。线性化的隐式反向欧拉法的块双对角结构使用并行循环消减进一步进行矢量化。在输入数据的两个轴上进行矢量化，提供了更高的计算带宽给计算设备，即使是相对稀疏数据的问题也能充分利用现代 GPU，实现了超过 100 倍的加速，与标准的顺序时间积分相比。我们通过几个示例问题，包括来自分析刚性和非刚性 ODE 模型以及神经 ODE 模型的问题，展示了隐式的矢量化时间积分的优势，并描述并提供了此处开发的方法的免费开源实现。

ODE 系统的时间向量化数值积分

Time-vectorized numerical integration for systems of ODEs

We present a novel, fast differentiable simulator for soft-body learning and
control applications. Existing differentiable soft-body simulators can be
classified into two categories based on their time integration methods:
Simulators using explicit time-stepping schemes require tiny time steps to
avoid numerical instabilities in gradient computation, and simulators using
implicit time integration typically compute gradients by employing the adjoint
method and solving the expensive linearized dynamics. Inspired by Projective
Dynamics (PD), we present Differentiable Projective Dynamics (DiffPD), an
efficient differentiable soft-body simulator based on PD with implicit time
integration. The key idea in DiffPD is to speed up backpropagation by
exploiting the prefactorized Cholesky decomposition in forward PD simulation.
In terms of contact handling, DiffPD supports two types of contacts: a
penalty-based model describing contact and friction forces and a
complementarity-based model enforcing non-penetration conditions and static
friction. We evaluate the performance of DiffPD and observe it is 4-19 times
faster compared with the standard Newton's method in various applications
including system identification, inverse design problems, trajectory
optimization, and closed-loop control. We also apply DiffPD in a
reality-to-simulation (real-to-sim) example with contact and collisions and
show its capability of reconstructing a digital twin of real-world scenes.

本文介绍了一种新型、快速可微的软体学习和控制应用模拟器 - Differentiable Projective Dynamics (DiffPD)，它以 Projective Dynamics (PD) 为基础具有隐式时间积分；DiffPD 利用向前 PD 模拟中的预因式化 Cholesky 分解加速反向传播，并在接触处理方面支持惩罚式模型和补偿式模型两种类型的接触。DiffPD 的表现非常优秀，可用于实现时域系统识别、逆向设计、轨迹优化和闭环控制应用。此外，利用 DiffPD 进行真实环境到仿真环境（real-to-sim）的数字化建模等方面的研究也取得了良好的成果。