Incorporating neural networks for the solution of Ordinary Differential
Equations (ODEs) represents a pivotal research direction within computational
mathematics. Within neural network architectures, the integration of the
intrinsic structure of ODEs offers advantages such as enhanced predictive
capabilities and reduced data utilization. Among these structural ODE forms,
the Lagrangian representation stands out due to its significant physical
underpinnings. Building upon this framework, Bhattoo introduced the concept of
Lagrangian Neural Networks (LNNs). Then in this article, we introduce a
groundbreaking extension (Genralized Lagrangian Neural Networks) to Lagrangian
Neural Networks (LNNs), innovatively tailoring them for non-conservative
systems. By leveraging the foundational importance of the Lagrangian within
Lagrange's equations, we formulate the model based on the generalized
Lagrange's equation. This modification not only enhances prediction accuracy
but also guarantees Lagrangian representation in non-conservative systems.
Furthermore, we perform various experiments, encompassing 1-dimensional and
2-dimensional examples, along with an examination of the impact of network
parameters, which proved the superiority of Generalized Lagrangian Neural
Networks(GLNNs).

基于广义拉格朗日方程的变形不仅提高了预测准确性，而且在非保守系统中保证了拉格朗日表示，从而证明了广义拉格朗日神经网络（GLNNs）的优越性。

广义拉格朗日神经网络

Generalized Lagrangian Neural Networks

Mathematical modeling with Ordinary Differential Equations (ODEs) has proven
to be extremely successful in a variety of fields, including biology. However,
these models are completely deterministic given a certain set of initial
conditions. We convert mathematical ODE models of three benchmark biological
systems to Dynamic Bayesian Networks (DBNs). The DBN model can handle model
uncertainty and data uncertainty in a principled manner. They can be used for
temporal data mining for noisy and missing variables. We apply Particle
Filtering algorithm to infer the model variables by re-estimating the models
parameters of various biological ODE models. The model parameters are
automatically re-estimated using temporal evidence in the form of data streams.
The results show that DBNs are capable of inferring the model variables of the
ODE model with high accuracy in situations where data is missing, incomplete,
sparse and irregular and true values of model parameters are not known.

本文介绍了利用 DBN 模型对三种生物 ODE 模型进行建模的过程，并使用粒子滤波算法对模型变量进行推断，其结果表明 DBNs 能够在缺失、不完整、稀疏和不规则的数据情况下，高精度地推断 ODE 模型的模型变量。