Machine and deep learning are reshaping physical simulation by enabling the development of surrogate models that either embed known constitutive laws or learn directly from high-fidelity simulation data generated by traditional numerical solvers. These data-driven and hybrid approaches offer new avenues for modeling, prediction, and control of complex systems— delivering faster, more efficient, and scalable alternatives to conventional computational methods. This mini-symposium explores recent advances at the intersection of machine learning and scientific computing for PDEs, and their applications. Topics of interest include deep learning and generative models for surrogate modeling, neural operators, reduced-order modeling, physics-informed and structure-preserving networks, data-driven discovery of system dynamics, and the mathematical foundations underpinning these techniques. We aim to bring together researchers exploring both theory and applications, working toward robust, generalizable, and physically consistent models for next-generation simulation tools in areas such as structural mechanics, fluid dynamics, materials modeling, and multi-physics simulation.