Classical numerical methods provide rigorous tools for simulating physical systems governed by partial differential equations; however, their computational cost grows rapidly with increasing system complexity, irregular geometries, and multi-scale or coupled dynamics. These limitations hinder their applicability in large-scale or real-time scenarios, such as predictive digital twins. To address these challenges, data-driven surrogate models have emerged, leveraging machine learning to approximate system responses directly from high-fidelity data, thereby enabling fast inference while retaining sufficient accuracy. Among these approaches, graph-based neural operators stand out as geometry-aware surrogates capable of handling complex and unstructured domains, and they constitute the core methodological foundation of this work. Despite their success, most geometric deep learning approaches rely on message-passing mechanisms, which suffer from fundamental limitations such as under-reaching [1], over-squashing, over-smoothing, and poor scalability. These issues stem from the intrinsic constraints of the message-passing paradigm and raise questions about whether graph-based representations are overly restrictive. Motivated by this, Topological Deep Learning has emerged as a powerful generalization, extending learning to richer topological structures such as simplicial complexes and hypergraphs [2]. This work explores how incorporating topological inductive biases can overcome the representational limits of standard MPNNs [3], enabling more expressive and scalable learning frameworks for the simulation of PDE-governed systems on meshes. In particular, the proposed approach enhances state-of-the-art methodologies by capturing complex dynamics arising from material heterogeneity, strong nonlinearity, and multiphysics interactions in coupled systems, where standard graph-based representations often fail to faithfully encode the underlying physical structure.