We present an end-to-end framework to learn partial differential equations
that brings together initial data production, selection of boundary conditions,
and the use of physics-informed neural operators to solve partial differential
equations that are ubiquitous in the study and modeling of physics phenomena.
We first demonstrate that our methods reproduce the accuracy and performance of
other neural operators published elsewhere in the literature to learn the 1D
wave equation and the 1D Burgers equation. Thereafter, we apply our
physics-informed neural operators to learn new types of equations, including
the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we
show that our approach is also applicable to learn the physics of the 2D linear
and nonlinear shallow water equations, which involve three coupled partial
differential equations. We release our artificial intelligence surrogates and
scientific software to produce initial data and boundary conditions to study a
broad range of physically motivated scenarios. We provide the source code, an
interactive website to visualize the predictions of our physics informed neural
operators, and a tutorial for their use at the Data and Learning Hub for
Science.

本文提出了一种端到端的框架，用于学习偏微分方程，首先通过物理感知神经算子实现了 1D 波动方程和 1D Burgers 方程的准确性和性能，并将其用于更多新的方程类型，以及在学习 2D 线性和非线性浅水方程中的应用。

物理启发的神经算子应用

Applications of physics informed neural operators

In this paper, we propose physics-informed neural operators (PINO) that uses
available data and/or physics constraints to learn the solution operator of a
family of parametric Partial Differential Equation (PDE). This hybrid approach
allows PINO to overcome the limitations of purely data-driven and physics-based
methods. For instance, data-driven methods fail to learn when data is of
limited quantity and/or quality, and physics-based approaches fail to optimize
on challenging PDE constraints. By combining both data and PDE constraints,
PINO overcomes all these challenges. Additionally, a unique property that PINO
enjoys over other hybrid learning methods is its ability to incorporate data
and PDE constraints at different resolutions. This allows us to combine
coarse-resolution data, which is inexpensive to obtain from numerical solvers,
with higher resolution PDE constraints, and the resulting PINO has no
degradation in accuracy even on high-resolution test instances. This
discretization-invariance property in PINO is due to neural-operator framework
which learns mappings between function spaces and allows evaluation at
different resolutions without the need for re-training. Moreover, PINO succeeds
in the purely physics setting, where no data is available, while other
approaches such as the Physics-Informed Neural Network (PINN) fail due to
optimization challenges, e.g. in multi-scale dynamic systems such as Kolmogorov
flows. This is because PINO learns the solution operator by optimizing PDE
constraints on multiple instances while PINN optimizes PDE constraints of a
single PDE instance. Further, in PINO, we incorporate the Fourier neural
operator (FNO) architecture which achieves orders-of-magnitude speedup over
numerical solvers and also allows us to compute explicit gradients on function
spaces efficiently.

本文提出物理信息神经操作器（PINO），该方法使用现有的数据和物理约束条件来学习参数化偏微分方程（PDE）族的解算器，通过结合数据和 PDE 约束条件，PINO 成功地实现了高分辨率实例的准确性和泛化能力。