The forward problems of pattern formation have been greatly empowered by extensive theoretical studies and simulations, however, the inverse problem is less well understood. It remains unclear how accurately one can use images of pattern formation to learn the functional forms of the nonlinear and nonlocal constitutive relations in the governing equation. We use PDE-constrained optimization to infer the governing dynamics and constitutive relations and use Bayesian inference and linearization to quantify their uncertainties in different systems, operating conditions, and imaging conditions. We discuss the conditions to reduce the uncertainty of the inferred functions and the correlation between them, such as state-dependent free energy and reaction kinetics (or diffusivity). We present the inversion algorithm and illustrate its robustness and uncertainties under limited spatiotemporal resolution, unknown boundary conditions, blurry initial conditions, and other non-ideal situations. Under certain situations, prior physical knowledge can be included to constrain the result. Phase-field, reaction-diffusion, and phase-field-crystal models are used as model systems. The approach developed here can find applications in inferring unknown physical properties of complex pattern-forming systems and in guiding their experimental design.

广泛的理论研究和模拟已极大地增强了图案形成的正向问题，但是，反问题还不太清楚。尚不清楚如何精确地使用图案形成的图像来学习控制方程中非线性和非局部本构关系的函数形式。我们使用PDE约束优化来推断控制动力学和本构关系，并使用贝叶斯推断和线性化来量化它们在不同系统，操作条件和成像条件下的不确定性。我们讨论了减少推断函数的不确定性以及它们之间的相关性的条件，例如取决于状态的自由能和反应动力学（或扩散性）。我们提出了反演算法，并说明了在有限时空分辨率，未知边界条件，模糊初始条件和其他非理想情况下的鲁棒性和不确定性。在某些情况下，可以包括先验的物理知识来约束结果。使用相场，反应扩散和相场晶体模型作为模型系统。此处开发的方法可以找到推断复杂图案形成系统未知物理特性并指导其实验设计的应用程序。相场晶体模型被用作模型系统。此处开发的方法可以找到推断复杂图案形成系统未知物理特性并指导其实验设计的应用程序。相场晶体模型被用作模型系统。此处开发的方法可以找到推断复杂图案形成系统未知物理特性并指导其实验设计的应用程序。