  One of the most promising applications of machine learning (ML) in
computational physics is to accelerate the solution of partial differential
equations (PDEs). The key objective of ML-based PDE solvers is to output a
sufficiently accurate solution faster than standard numerical methods, which
are used as a baseline comparison. We first perform a systematic review of the
ML-for-PDE solving literature. Of articles that use ML to solve a fluid-related
PDE and claim to outperform a standard numerical method, we determine that 79%
(60/76) compare to a weak baseline. Second, we find evidence that reporting
biases, especially outcome reporting bias and publication bias, are widespread.
We conclude that ML-for-PDE solving research is overoptimistic: weak baselines
lead to overly positive results, while reporting biases lead to underreporting
of negative results. To a large extent, these issues appear to be caused by
factors similar to those of past reproducibility crises: researcher degrees of
freedom and a bias towards positive results. We call for bottom-up cultural
changes to minimize biased reporting as well as top-down structural reforms
intended to reduce perverse incentives for doing so.
