Graph Neural Networks (GNNs) are emerging as powerful tools for nonlinear Model Order Reduction (MOR) of time-dependent parameterized Partial Differential Equations (PDEs). However, existing methodologies struggle to combine geometric inductive biases with interpretable latent behavior, overlooking dynamics-driven features or disregarding spatial information. In this work, we address this gap by introducing Latent Dynamics Graph Convolutional Network (LD-GCN), a purely data-driven, encoder-free architecture that learns a global, low-dimensional representation of dynamical systems conditioned on external inputs and parameters. The temporal evolution is modeled in the latent space and advanced through time-stepping, allowing for time-extrapolation, and the trajectories are consistently decoded onto geometrically parameterized domains using a GNN. Our framework enhances interpretability by enabling the analysis of the reduced dynamics and supporting zero-shot prediction through latent interpolation. The methodology is mathematically validated via a universal approximation theorem for encoder-free architectures, and numerically tested on complex computational mechanics problems involving physical and geometric parameters, including the detection of bifurcating phenomena for Navier-Stokes equations. Code availability: https://github.com/lorenzotomada/ld-gcn-rom

翻译：图神经网络（GNNs）正日益成为处理时变参数化偏微分方程（PDEs）非线性模型降阶（MOR）的强大工具。然而，现有方法难以将几何归纳偏置与可解释的潜在行为相结合，往往忽略了动力学驱动的特征或忽视了空间信息。本文通过提出潜在动力学图卷积网络（LD-GCN）来填补这一空白，该架构为纯数据驱动、无编码器的结构，能够学习以外部输入和参数为条件的动力系统的全局低维表示。时间演化在潜在空间中进行建模，并通过时间步进推进，从而实现时间外推；轨迹则通过GNN一致地解码到几何参数化的域上。我们的框架通过支持对降阶动力学的分析以及借助潜在插值实现零样本预测，增强了模型的可解释性。该方法通过无编码器架构的通用逼近定理进行了数学验证，并在涉及物理和几何参数的复杂计算力学问题上进行了数值测试，包括对Navier-Stokes方程分岔现象的检测。代码可用性：https://github.com/lorenzotomada/ld-gcn-rom