Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization routines, model-based control, or solution of large-scale inverse problems. Existing Convolutional Neural Network-based frameworks for surrogate modeling require lossy pixelization and data-preprocessing, which is not suitable for realistic engineering applications. Therefore, we propose non-linear independent dual system (NIDS), which is a deep learning surrogate model for discretization-independent, continuous representation of PDE solutions, and can be used for prediction over domains with complex, variable geometries and mesh topologies. NIDS leverages implicit neural representations to develop a non-linear mapping between problem parameters and spatial coordinates to state predictions by combining evaluations of a case-wise parameter network and a point-wise spatial network in a linear output layer. The input features of the spatial network include physical coordinates augmented by a minimum distance function evaluation to implicitly encode the problem geometry. The form of the overall output layer induces a dual system, where each term in the map is non-linear and independent. Further, we propose a minimum distance function-driven weighted sum of NIDS models using a shared parameter network to enforce boundary conditions by construction under certain restrictions. The framework is applied to predict solutions around complex, parametrically-defined geometries on non-parametrically-defined meshes with solutions obtained many orders of magnitude faster than the full order models. Test cases include a vehicle aerodynamics problem with complex geometry and data scarcity, enabled by a training method in which more cases are gradually added as training progresses.
翻译:部分差异方程式(PDEs)的数值解决方案需要昂贵的模拟,限制了其在设计优化常规、模型基控制或大规模反向问题解决方案中的应用。现有的代用模型的演进神经网络现有框架需要丢失像素化和数据预处理,这不适合现实工程应用。因此,我们提议非线性独立双向系统(NIDS),这是离散独立、持续代表PDE解决方案的深度学习替代模型,可用于在复杂、可变的地貌和米色表层的域上预测。NIDS利用隐含的神经显示来开发问题参数和空间坐标之间的非线性绘图,以便通过综合对案例参数网络的评价和线性输出层的点向空间网络网络进行预测。空间网络的输入特征包括物理协调,通过最小的距离函数来强化问题定义的地理测量。总体输出层的形式是双向系统,其中每个术语都是非线性地理分布式和网状表的,可以逐步地在系统内应用一个不线性和独立的模型。我们提出一个最小的远程计算方法,用来在结构结构框架下执行某种测算式的系统。我们提出一个最小的计算方法,用一个比较精确的模型来测量式的系统,在系统里测算中,用一个总的模型进行测测算,用一个比数式的系统里程的模型来测量式的计算,用一个测量式的计算,用一个总路路路路路路路面的模型,用一个总路路路路路路路路路路路路路路路路路路系的模型,用一个比。