<p dir="ltr">The integration of scientific machine learning with computational structural mechanics offers a range of opportunities to address some of the most significant challenges currently experienced by multiphysical simulations, design optimization, and inverse sensing problems. While traditional mesh-based numerical methods, such as the Finite Element Method (FEM), have proven to be very powerful when applied to complex and geometrically inhomogeneous domains, their performance deteriorates very rapidly when faced with simulation scenarios involving high-dimensional systems, high-frequency inputs and outputs, and highly irregular domains. All these elements contribute to increase in the overall computational cost, the mesh dependence, and the number of costly matrix operations that can rapidly render FEM inapplicable. In a similar way, traditional inverse solvers, including global optimization methods, also face important limitations when handling high-dimensional, dynamic design spaces, and multiphysics systems. Recent advances in machine learning (ML) and deep learning have opened new ways to develop alternative techniques for the simulation of complex engineering systems. However, most of the existing deep learning methods are data greedy, a property that strides with the typically limited availability of physical observations and data in scientific applications. This sharp contrast between needed and available data can lead to poor approximations and physically inconsistent solutions. An opportunity to overcome this problem is offered by the class of so-called physics-informed or scientific machine learning methods that leverage the knowledge of problem-specific governing physics to alleviate, or even completely eliminate, the dependence on data. As a result, this class of methods can leverage the advantages of ML algorithms without inheriting their data greediness. This dissertation aims to develop scientific ML methods for application to forward and inverse problems in acoustics and structural mechanics while simultaneously overcoming some of the most significant limitations of traditional computational mechanics methods. </p><p dir="ltr">This work develops fully physics-driven deep learning frameworks specifically conceived to perform forward <i>simulations</i> of mechanical systems that provide approximate, yet physically consistent, solutions without requiring labeled data. The proposed set of approaches is characterized by low discretization dependence and is conceived to support parallel computations in future developments. These characteristics make these methods efficient to handle high degrees of freedom systems, high-frequency simulations, and systems with irregular geometries. The proposed deep learning frameworks enforce the governing equations within the deep learning algorithm, therefore removing the need for costly training data generation while preserving the physical accuracy of the simulation results. Another noteworthy contribution consists in the development of a fully physics-driven deep learning framework capable of improving the computational time for simulating domains with irregular geometries by orders of magnitude in comparison to the traditional mesh-based methods. This novel framework is both geometry-aware and maintains physical consistency throughout the simulation process. The proposed framework displays the remarkable ability to simulate systems with different domain geometries without the need for a new model assembly or a training phase. This capability is in stark contrast with current numerical mesh-based methods, that require new model assembly, and with conventional ML models, that require new training.</p><p dir="ltr">In the second part of this dissertation, the work focuses on the development of ML-based approaches to solve inverse problems. A new deep reinforcement learning framework tailored for dynamic <i>design optimization</i> tasks in coupled-physics problems is presented. The framework effectively addresses key limitations of traditional methods by enabling the exploration of high-dimensional design spaces and supporting sequential decision-making in complex multiphysics systems. Maintaining the focus on the class of inverse problems, ML-based algorithms for <i>remote sensing</i> are also explored with particular reference to structural health monitoring applications. A modular neural network framework is formulated by integrating three essential modules: physics-based regularization, geometry-based regularization, and reduced-order representation. The concurrent use of these modules has shown remarkable performance when addressing the challenges associated with nonlinear, high-dimensional, and often ill-posed remote sensing problems. Finally, this dissertation illustrates the efficacy of deep learning approaches for experimental remote sensing. Results show the significant ability of these techniques when applied to learning inverse mappings based on high-dimensional and noisy experimental data. The proposed framework incorporates data augmentation and denoising techniques to handle limited and noisy experimental datasets, hence establishing a robust approach for training on experimental data.</p>