A critical issue in approximating solutions of ordinary differential
equations using neural networks is the exact satisfaction of the boundary or
initial conditions. For this purpose, neural forms have been introduced, i.e.,
functional expressions that depend on neural networks which, by design, satisfy
the prescribed conditions exactly. Expanding upon prior progress, the present
work contributes in three distinct aspects. First, it presents a novel
formalism for crafting optimized neural forms. Second, it outlines a method for
establishing an upper bound on the absolute deviation from the exact solution.
Third, it introduces a technique for converting problems with Neumann or Robin
conditions into equivalent problems with parametric Dirichlet conditions. The
proposed optimized neural forms were numerically tested on a set of diverse
problems, encompassing first-order and second-order ordinary differential
equations, as well as first-order systems. Stiff and delay differential
equations were also considered. The obtained solutions were compared against
solutions obtained via Runge-Kutta methods and exact solutions wherever
available. The reported results and analysis verify that in addition to the
exact satisfaction of the boundary or initial conditions, optimized neural
forms provide closed-form solutions of superior interpolation capability and
controllable overall accuracy.

使用神经网络近似求解常微分方程时，关键问题是确切满足边界条件或初始条件。本文提出了一种优化神经形式的新形式，并提出了一种建立与精确解的绝对偏差的上界的方法。同时，还介绍了将具有 Neumann 或 Robin 条件的问题转化为带参数的 Dirichlet 条件的等价问题的技术。通过对一组不同问题进行数值测试，结果显示优化的神经形式不仅能够精确满足边界和初始条件，还提供具有优越插值能力和可控整体精度的闭式解。

用于解决常微分方程的优化神经形式

Optimized neural forms for solving ordinary differential equations

Current physics-informed (standard or operator) neural networks still rely on
accurately learning the initial conditions of the system they are solving. In
contrast, standard numerical methods evolve such initial conditions without
needing to learn these. In this study, we propose to improve current
physics-informed deep learning strategies such that initial conditions do not
need to be learned and are represented exactly in the predicted solution.
Moreover, this method guarantees that when a DeepONet is applied multiple times
to time step a solution, the resulting function is continuous.

通过改进当前的物理信息神经网络深度学习策略，本研究提出了不需要学习初始条件且以精确预测解决方案为代表的方法。此外，该方法确保当多次应用 DeepONet 以进行解决方案的时间步进时，得到的函数是连续的。

改进具有硬限制条件的物理信息化 DeepONets

Improving physics-informed DeepONets with hard constraints

The standard procedure to generate initial conditions (IC) in numerical
simulations is to use the Zel'dovich approximation (ZA). Although the ZA
correctly reproduces the linear growing modes of density and velocity
perturbations, non-linear growth is inaccurately represented because of the ZA
failure to conserve momentum. This implies that it takes time for the actual
dynamics to establish the correct statistical properties of density and
velocity fields. We extend perturbation theory (PT) to include transients as
non-linear excitations of decaying modes caused by the IC. We focus on
higher-order statistics of the density contrast and velocity divergence,
characterized by the S_p and T_p parameters. We find that the time-scale of
transients is determined, at a given order p, by the spectral index n. The
skewness factor S_3 (T_3) attains 10% accuracy only after a=6 (a=15) for n=0,
whereas higher (lower) n demands more (less) expansion away from the IC. These
requirements become much more stringent as p increases. An Omega=0.3 model
requires a factor of two larger expansion than an Omega=1 model to reduce
transients by the same amount. The predicted transients in S_p are in good
agreement with numerical simulations. More accurate IC can be achieved by using
2nd order Lagrangian PT (2LPT), which reproduces growing modes up to 2nd order
and thus eliminates transients in the skewness. We show that for p>3 this
reduces the required expansion by more than an order of magnitude compared to
the ZA. Setting up 2LPT IC only requires minimal, inexpensive changes to ZA
codes. We suggest simple steps for its implementation.

研究了在使用 Zel'dovich 近似初始条件生成模拟的模拟中，考虑到高阶统计量时，使用 2 级拉格朗日扰动理论生成的初始条件能够比 Zel'dovich 近似生成的初始条件更准确地预测转瞬即逝的模拟现象，并提出了一种实现 2 级拉格朗日扰动理论生成初始条件的简单步骤。