Throughout many fields, practitioners often rely on differential equations to
model systems. Yet, for many applications, the theoretical derivation of such
equations and/or accurate resolution of their solutions may be intractable.
Instead, recently developed methods, including those based on parameter
estimation, operator subset selection, and neural networks, allow for the
data-driven discovery of both ordinary and partial differential equations
(PDEs), on a spectrum of interpretability. The success of these strategies is
often contingent upon the correct identification of representative equations
from noisy observations of state variables and, as importantly and intertwined
with that, the mathematical strategies utilized to enforce those equations.
Specifically, the latter has been commonly addressed via unconstrained
optimization strategies. Representing the PDE as a neural network, we propose
to discover the PDE by solving a constrained optimization problem and using an
intermediate state representation similar to a Physics-Informed Neural Network
(PINN). The objective function of this constrained optimization problem
promotes matching the data, while the constraints require that the PDE is
satisfied at several spatial collocation points. We present a penalty method
and a widely used trust-region barrier method to solve this constrained
optimization problem, and we compare these methods on numerical examples. Our
results on the Burgers' and the Korteweg-De Vreis equations demonstrate that
the latter constrained method outperforms the penalty method, particularly for
higher noise levels or fewer collocation points. For both methods, we solve
these discovered neural network PDEs with classical methods, such as finite
difference methods, as opposed to PINNs-type methods relying on automatic
differentiation. We briefly highlight other small, yet crucial, implementation
details.

通过解决受约束的优化问题并使用类似于物理 - Informed 神经网络（PINN）的中间状态表示，我们将 PDE 表示为神经网络，以发现 PDE。我们使用惩罚方法和广泛使用的约束 - 区域障碍方法解决了此约束优化问题，并在数值示例上比较了这些方法。我们对 Burgers' 和 Korteweg-De Vreis 方程的结果表明，后一种约束方法在更高的噪声水平或更少的空间插值点上表现优于惩罚方法。对于这两种方法，我们使用传统方法（如有限差分方法）解决这些发现的神经网络 PDE，而不是依赖于自动微分的 PINNs 类型方法。我们简要介绍其他一些小但至关重要的实施细节。

从数据中基于神经网络的方程发现：约束还是无约束？

Constrained or Unconstrained? Neural-Network-Based Equation Discovery  from Data

The solution of a partial differential equation can be obtained by computing
the inverse operator map between the input and the solution space. Towards this
end, we introduce a \textit{multiwavelet-based neural operator learning scheme}
that compresses the associated operator's kernel using fine-grained wavelets.
By explicitly embedding the inverse multiwavelet filters, we learn the
projection of the kernel onto fixed multiwavelet polynomial bases. The
projected kernel is trained at multiple scales derived from using repeated
computation of multiwavelet transform. This allows learning the complex
dependencies at various scales and results in a resolution-independent scheme.
Compare to the prior works, we exploit the fundamental properties of the
operator's kernel which enable numerically efficient representation. We perform
experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy
Flow, and Navier-Stokes equation. Compared with the existing neural operator
approaches, our model shows significantly higher accuracy and achieves
state-of-the-art in a range of datasets. For the time-varying equations, the
proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative
$L2$ error for Burgers' (KdV) equation). By learning the mappings between
function spaces, the proposed method has the ability to find the solution of a
high-resolution input after learning from lower-resolution data.

本文介绍了一种基于多小波神经算子学习方案的解微分方程的方法，该方法使用细粒度小波压缩相关算子的内核，并通过显式嵌入逆小波滤波器，将内核投影到固定的多项式基上进行训练，在多个尺度上训练投影内核，实现了解微分方程的分辨率独立方案。与现有的神经算子方法相比，我们的模型在一系列数据集上展现出更高的准确性，并取得了最新的成果。