While deep learning techniques have become extremely popular for solving a broad range of optimization problems, methods to enforce hard constraints during optimization, particularly on deep neural networks, remain underdeveloped. Inspired by the rich literature on meshless interpolation and its extension to spectral collocation methods in scientific computing, we develop a series of approaches for enforcing hard constraints on neural fields, which we refer to as \emph{Constrained Neural Fields} (CNF). The constraints can be specified as a linear operator applied to the neural field and its derivatives. We also design specific model representations and training strategies for problems where standard models may encounter difficulties, such as conditioning of the system, memory consumption, and capacity of the network when being constrained. Our approaches are demonstrated in a wide range of real-world applications. Additionally, we develop a framework that enables highly efficient model and constraint specification, which can be readily applied to any downstream task where hard constraints need to be explicitly satisfied during optimization.

翻译：尽管深度学习技术已在解决广泛的优化问题中变得极为流行，但在优化过程中强制执行硬约束（尤其是针对深度神经网络）的方法仍发展不足。受科学计算中无网格插值及其在谱配点方法中扩展的丰富文献启发，我们开发了一系列对神经场施加硬约束的方法，称之为"约束神经场"（Constrained Neural Fields, CNF）。这些约束可表示为作用于神经场及其导数的线性算子。针对标准模型可能遇到困难的问题（例如系统条件性、内存消耗以及受约束时网络的容量），我们还设计了特定的模型表示和训练策略。我们的方法在多种实际应用中得到验证。此外，我们开发了一个支持高效模型与约束规范的框架，可轻松应用于任何需要在优化过程中显式满足硬约束的下游任务。