In the era of AI, neural networks have become increasingly popular for modeling, inference, and prediction, largely due to their potential for universal approximation. With the proliferation of such deep learning models, a question arises: are leaner statistical methods still relevant? To shed insight on this question, we employ the mechanistic nonlinear ordinary differential equation (ODE) inverse problem as a testbed, using the physics-informed neural network (PINN) as a representative of the deep learning paradigm and manifold-constrained Gaussian process inference (MAGI) as a representative of statistically principled methods. Through case studies involving the SEIR model from epidemiology and the Lorenz model from chaotic dynamics, we demonstrate that statistical methods are far from obsolete, especially when working with sparse and noisy observations. On tasks such as parameter inference and trajectory reconstruction, statistically principled methods consistently achieve lower bias and variance, while using far fewer parameters and requiring less hyperparameter tuning. Statistical methods can also decisively outperform deep learning models on out-of-sample future prediction, where the absence of relevant data often leads overparameterized models astray. Additionally, we find that statistically principled approaches are more robust to accumulation of numerical imprecision and can represent the underlying system more faithfully to the true governing ODEs.

翻译：在人工智能时代，神经网络因其通用逼近潜力而在建模、推断和预测领域日益普及。随着此类深度学习模型的大量涌现，一个问题随之产生：更为精简的统计方法是否仍具有价值？为探究这一问题，我们以机械性非线性常微分方程逆问题为试验平台，采用物理信息神经网络作为深度学习范式的代表，以及流形约束高斯过程推断作为统计原理方法的代表。通过涉及流行病学SEIR模型和混沌动力学Lorenz模型的案例研究，我们证明统计方法远未过时，特别是在处理稀疏且含噪声观测数据时。在参数推断和轨迹重建等任务中，基于统计原理的方法持续实现更低的偏差和方差，同时使用的参数更少且超参数调优需求更低。在样本外未来预测任务中，统计方法也能显著优于深度学习模型——后者因缺乏相关数据常导致过参数化模型偏离正确方向。此外，我们发现基于统计原理的方法对数值精度累积误差具有更强鲁棒性，并能更忠实地表征与真实控制常微分方程对应的底层系统。