Often, surrogate modeling is conducted purely based on large amounts of data, mostly by training ANNs with them. However, within the context of continuum (micro-)mechanics, there exists a wealth of established physical and empirical knowledge that ANN-based surrogate methods have yet to fully incorporate. In the following, we discuss the notable challenges in bridging this gap for surrogate modeling in continuum mechanics. a) Physics-enhanced surrogate modeling: Incorporating physics-based knowledge into surrogate modeling is an active research field. For instance, in and , physics-based knowledge, including the underlying PDEs and empirical knowledge, has been leveraged to introduce biases into ANNs, resulting in outputs that approximate the underlying physics, such as enforcing divergence-free conditions as well as mass and energy conservation. However, it is essential to note that these approaches primarily aim to satisfy the physical laws in a weak sense. Hence, the output from the trained ANNs may not be fully physically meaningful, particularly at a local scale. Therefore, we need to explore the design of neural network architectures that are capable to inherently produce an output that satisfies physics in a strong sense, with a particular focus on critical properties like divergence-free behavior, as well as mass and energy conservation both, on a global and local scale. b) Stable dynamic prediction: Surrogate modeling has been used for predicting time-dependent stress and stress fields of heterogeneous solids subject to homogenous steady-state external loading conditions. Within this framework, these surrogate models can be regarded as dynamical systems. Given that surrogate models typically emulate stable physical behavior thereby mimicking the basic rules of continuum mechanics, it is essential for them to possess inherent stability, i.e. mimicking also convergence. This stability ensures that predictions remain consistently stable and bounded. Consequently, it is imperative that ANN-based surrogate models are designed to have these stability properties inherently embedded. c) Learning low-dimensional latent representation: Often, the field of interest in continuum mechanics is two or three-dimensional real space, ideally also informed by the solid's crystal and phase state, adding further dimensions and anisotropy features to the problem to be solved. Consequently, the data obtained for these scenarios are high-dimensional, especially while dealing with high-resolution spatial fields. However, it is a common observation that such highdimensional solutions can often be accurately represented in a low-dimensional latent space. The creation of this low-dimensional space is further guided by constraints designed to simplify the dynamics and engineering design processes. For instance, it is possible to construct a latent space in such a way that the system dynamics evolve in a nearly linear fashion, aligning with principles like Koopman theory and dimensionality reduction techniques.