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.

翻译：将物理知识嵌入神经网络训练一直是研究热点。然而，面对复杂的现实世界，现有方法大多仍然强烈依赖观测数据的数量和质量。此外，当真实方程的解非常复杂时，神经网络往往难以收敛。受计算流体动力学中大涡模拟的启发，我们提出了一种基于滤波的改进方法。我们分析了物理信息机器学习中困难的原因，并提出了一种原始物理方程的代理约束（简称滤波PDE，FPDE），以降低噪声和稀疏观测数据的影响。在噪声与稀疏性实验中，所提出的FPDE模型（通过FPDE约束优化）比传统PDE模型具有更好的鲁棒性。实验表明，FPDE模型能够在基线模型观测数据量12%且噪声水平高出100%的条件下获得同等质量的解。此外，使用两组真实测量数据展示了FPDE在实际案例中的改进效果。最终结果表明，在面对不完整方程问题以及极端稀疏和高噪声条件时，FPDE仍能给出物理上更合理的解。对于将真实实验数据融入物理信息训练，所提出的FPDE约束在两项真实世界实验中表现良好：血管内血流速度建模和划痕中细胞迁移建模。