We revisit the problem of physics-informed regression, and propose a method that directly computes the state at the prediction point, simultaneously with the derivative and curvature information of the existing samples. We frame each prediction as a constrained optimisation problem, leveraging multivariate Taylor series expansions and explicitly enforcing physical laws. Each individual query can be processed with low computational cost without any pre- or re-training, in contrast to global function approximator-based solutions such as neural networks. Our comparative benchmarks on a reaction-diffusion system show competitive predictive accuracy relative to a neural network-based solution, while completely eliminating the need for long training loops, and remaining robust to changes in the sampling layout.

翻译：本文重新探讨物理信息回归问题，提出一种直接计算预测点状态的方法，同时结合现有样本的导数与曲率信息。我们将每个预测构建为约束优化问题，利用多元泰勒级数展开并显式强制物理定律约束。与基于全局函数逼近器（如神经网络）的解决方案相比，该方法无需任何预训练或重新训练即可以较低计算成本处理单个查询。在反应-扩散系统上的对比实验表明，相较于基于神经网络的解决方案，本方法在保持预测精度的同时，完全消除了长训练循环的需求，并对采样布局的变化保持鲁棒性。