Recent work provides promising evidence that Physics-informed neural networks
(PINN) can efficiently solve partial differential equations (PDE). However,
previous works have failed to provide guarantees on the worst-case residual
error of a PINN across the spatio-temporal domain - a measure akin to the
tolerance of numerical solvers - focusing instead on point-wise comparisons
between their solution and the ones obtained by a solver on a set of inputs. In
real-world applications, one cannot consider tests on a finite set of points to
be sufficient grounds for deployment, as the performance could be substantially
worse on a different set. To alleviate this issue, we establish tolerance-based
correctness conditions for PINNs over the entire input domain. To verify the
extent to which they hold, we introduce $\partial$-CROWN: a general, efficient
and scalable post-training framework to bound PINN residual errors. We
demonstrate its effectiveness in obtaining tight certificates by applying it to
two classically studied PDEs - Burgers' and Schr\"odinger's equations -, and
two more challenging ones with real-world applications - the Allan-Cahn and
Diffusion-Sorption equations.

本文介绍了一种基于物理信息的神经网络（PINN）来解决偏微分方程的方法，并提出了一种基于容差的正确性条件的后训练框架（CROWN），用于限制 PINN 残差误差，并在经典 PDE 和现实应用中进行了实际效果测试。

可证明正确的物理信息神经网络

Provably Correct Physics-Informed Neural Networks

We give the first dimensionality reduction methods for the overconstrained
Tukey regression problem. The Tukey loss function $\|y\|_M = \sum_i M(y_i)$ has
$M(y_i) \approx |y_i|^p$ for residual errors $y_i$ smaller than a prescribed
threshold $\tau$, but $M(y_i)$ becomes constant for errors $|y_i| > \tau$. Our
results depend on a new structural result, proven constructively, showing that
for any $d$-dimensional subspace $L \subset \mathbb{R}^n$, there is a fixed
bounded-size subset of coordinates containing, for every $y \in L$, all the
large coordinates, with respect to the Tukey loss function, of $y$. Our methods
reduce a given Tukey regression problem to a smaller weighted version, whose
solution is a provably good approximate solution to the original problem. Our
reductions are fast, simple and easy to implement, and we give empirical
results demonstrating their practicality, using existing heuristic solvers for
the small versions. We also give exponential-time algorithms giving provably
good solutions, and hardness results suggesting that a significant speedup in
the worst case is unlikely.

本文提出了用于过约束 Tukey 回归问题的首个降维方法，并使用新的结构结果来证明结果。作者的方法简单易行且易于实现，对于小版本的问题我们使用现有的启发式求解器来证明它们的实用性，同时指出在最坏情况下极少有可能获得显着的速度提升。

Tukey 回归的降维

Dimensionality Reduction for Tukey Regression

Multivariate time-series modeling and forecasting is an important problem
with numerous applications. Traditional approaches such as VAR (vector
auto-regressive) models and more recent approaches such as RNNs (recurrent
neural networks) are indispensable tools in modeling time-series data. In many
multivariate time series modeling problems, there is usually a significant
linear dependency component, for which VARs are suitable, and a nonlinear
component, for which RNNs are suitable. Modeling such times series with only
VAR or only RNNs can lead to poor predictive performance or complex models with
large training times. In this work, we propose a hybrid model called R2N2
(Residual RNN), which first models the time series with a simple linear model
(like VAR) and then models its residual errors using RNNs. R2N2s can be trained
using existing algorithms for VARs and RNNs. Through an extensive empirical
evaluation on two real world datasets (aviation and climate domains), we show
that R2N2 is competitive, usually better than VAR or RNN, used alone. We also
show that R2N2 is faster to train as compared to an RNN, while requiring less
number of hidden units.

提出了一种名为 R2N2（残余 RNN）的混合模型，通过简单的线性模型（如 VAR）对时间序列进行建模，并使用 RNN 对其残余误差进行建模，在航空领域和气候领域中的实证评估表明，与单独使用 VAR 或 RNN 相比，R2N2 具有竞争力且更快速地训练，同时需要较少数量的隐藏单元。