Machine learning and neural networks have advanced numerous research domains, but challenges such as large training data requirements and inconsistent model performance hinder their application in certain scientific problems. To overcome these challenges, researchers have investigated integrating physics principles into machine learning models, mainly through: (i) physics-guided loss functions, generally termed as physics-informed neural networks, and (ii) physics-guided architectural design. While both approaches have demonstrated success across multiple scientific disciplines, they have limitations including being trapped to a local minimum, poor interpretability, and restricted generalizability. This paper proposes a new physics-informed neural network (PINN) architecture that combines the strengths of both approaches by embedding the fundamental solution of the wave equation into the network architecture, enabling the learned model to strictly satisfy the wave equation. The proposed point neuron learning method can model an arbitrary sound field based on microphone observations without any dataset. Compared to other PINN methods, our approach directly processes complex numbers and offers better interpretability and generalizability. We evaluate the versatility of the proposed architecture by a sound field reconstruction problem in a reverberant environment. Results indicate that the point neuron method outperforms two competing methods and can efficiently handle noisy environments with sparse microphone observations.

翻译：机器学习和神经网络已推动众多研究领域的发展，但在某些科学问题中的应用仍受限于训练数据需求大、模型性能不稳定等挑战。为克服这些挑战，研究者探索了将物理原理整合到机器学习模型中的方法，主要包括：（i）物理引导的损失函数（通常称为物理信息神经网络），以及（ii）物理引导的架构设计。尽管这两种方法在多个科学领域已取得成功，但仍存在局部最优解陷阱、可解释性差和泛化能力受限等不足。本文提出一种新的物理信息神经网络架构，通过将波动方程的基本解嵌入网络结构，结合了上述两种方法的优势，使学习模型能严格满足波动方程。所提出的点神经元学习方法可基于麦克风观测数据建模任意声场，无需任何数据集。与其他物理信息神经网络方法相比，该方法直接处理复数，并具有更好的可解释性与泛化能力。我们通过混响环境中的声场重建问题评估了该架构的通用性。结果表明，点神经元方法优于两种对比方法，并能有效处理稀疏麦克风观测下的噪声环境。