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

Filtered Partial Differential Equations: a robust surrogate constraint in physics-informed deep learning framework