We introduce the concept of geometry-informed neural networks (GINNs), which encompass (i) learning under geometric constraints, (ii) neural fields as a suitable representation, and (iii) generating diverse solutions to under-determined systems often encountered in geometric tasks. Notably, the GINN formulation does not require training data, and as such can be considered generative modeling driven purely by constraints. We add an explicit diversity loss to mitigate mode collapse. We consider several constraints, in particular, the connectedness of components which we convert to a differentiable loss through Morse theory. Experimentally, we demonstrate the efficacy of the GINN learning paradigm across a range of two and three-dimensional scenarios with increasing levels of complexity.

翻译：我们提出几何信息神经网络（GINNs）的概念，该框架包含三大核心要素：（i）在几何约束下进行学习；（ii）以神经场作为合适的表征形式；（iii）为几何任务中常见的欠定系统生成多样化解。值得注意的是，GINN公式无需训练数据，因此可视为纯粹由约束驱动的生成式建模。我们引入显式多样性损失函数以缓解模式坍塌现象。针对多种约束条件，特别是通过莫尔斯理论将组件的连通性约束转化为可微损失函数。实验表明，GINN学习范式在二维与三维场景中均展现出显著效能，且能有效应对从简单到复杂的多层次挑战。