  Embedding physical knowledge into neural network (NN) training has been a hot
topic. However, when facing the complex real-world, most of the existing
methods still strongly rely on the quantity and quality of observation data.
Furthermore, the neural networks often struggle to converge when the solution
to the real equation is very complex. Inspired by large eddy simulation in
computational fluid dynamics, we propose an improved method based on filtering.
We analyzed the causes of the difficulties in physics informed machine
learning, and proposed a surrogate constraint (filtered PDE, FPDE in short) of
the original physical equations to reduce the influence of noisy and sparse
observation data. In the noise and sparsity experiment, the proposed FPDE
models (which are optimized by FPDE constraints) have better robustness than
the conventional PDE models. Experiments demonstrate that the FPDE model can
obtain the same quality solution with 100% higher noise and 12% quantity of
observation data of the baseline. Besides, two groups of real measurement data
are used to show the FPDE improvements in real cases. The final results show
that FPDE still gives more physically reasonable solutions when facing the
incomplete equation problem and the extremely sparse and high-noise conditions.
For combining real-world experiment data into physics-informed training, the
proposed FPDE constraint is useful and performs well in two real-world
experiments: modeling the blood velocity in vessels and cell migration in
scratches.
