Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations
This addresses a critical bottleneck in data-driven PDE discovery for fields like physics and finance, enabling more accurate modeling of complex natural phenomena from scarce or noisy data.
The paper tackles the challenge of discovering highly nonlinear parametric partial differential equations (PDEs) from low-quality data by proposing a physics-guided learning method, resulting in reduced estimation errors, robustness against noise, and the first successful discovery of three-dimensional PDEs with highly nonlinear coefficients.
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from scientific data thrives as a new attempt to model complex phenomena in nature, but the effectiveness of current practice is typically limited by the scarcity of data and the complexity of phenomena. Especially, the discovery of PDEs with highly nonlinear coefficients from low-quality data remains largely under-addressed. To deal with this challenge, we propose a novel physics-guided learning method, which can not only encode observation knowledge such as initial and boundary conditions but also incorporate the basic physical principles and laws to guide the model optimization. We theoretically show that our proposed method strictly reduces the coefficient estimation error of existing baselines, and is also robust against noise. Extensive experiments show that the proposed method is more robust against data noise, and can reduce the estimation error by a large margin. Moreover, all the PDEs in the experiments are correctly discovered, and for the first time we are able to discover three-dimensional PDEs with highly nonlinear coefficients.