Despite incredible progress, many neural architectures fail to properly generalize beyond their training distribution. As such, learning to reason in a correct and generalizable way is one of the current fundamental challenges in machine learning. In this respect, logic puzzles provide a great testbed, as we can fully understand and control the learning environment. Thus, they allow to evaluate performance on previously unseen, larger and more difficult puzzles that follow the same underlying rules. Since traditional approaches often struggle to represent such scalable logical structures, we propose to model these puzzles using a graph-based approach. Then, we investigate the key factors enabling the proposed models to learn generalizable solutions in a reinforcement learning setting. Our study focuses on the impact of the inductive bias of the architecture, different reward systems and the role of recurrent modeling in enabling sequential reasoning. Through extensive experiments, we demonstrate how these elements contribute to successful extrapolation on increasingly complex puzzles.These insights and frameworks offer a systematic way to design learning-based systems capable of generalizable reasoning beyond interpolation.

In this work, we devise a new, general-purpose reinforcement learning strategy for the optimal control of parametric partial differential equations (PDEs). Such problems frequently arise in applied sciences and engineering and entail a significant complexity when control and/or state variables are distributed in high-dimensional space or depend on varying parameters. Traditional numerical methods, relying on either iterative minimization algorithms or dynamic programming, while reliable, often become computationally infeasible. Indeed, in either way, the optimal control problem must be solved for each instance of the parameters, and this is out of reach when dealing with high-dimensional time-dependent and parametric PDEs. In this paper, we propose HypeRL, a deep reinforcement learning (DRL) framework to overcome the limitations shown by traditional methods. HypeRL aims at approximating the optimal control policy directly. Specifically, we employ an actor-critic DRL approach to learn an optimal feedback control strategy that can generalize across the range of variation of the parameters. To effectively learn such optimal control laws, encoding the parameter information into the DRL policy and value function neural networks (NNs) is essential. To do so, HypeRL uses two additional NNs, often called hypernetworks, to learn the weights and biases of the value function and the policy NNs. We validate the proposed approach on two PDE-constrained optimal control benchmarks, namely a 1D Kuramoto-Sivashinsky equation and a 2D Navier-Stokes equations, by showing that the knowledge of the PDE parameters and how this information is encoded, i.e., via a hypernetwork, is an essential ingredient for learning parameter-dependent control policies that can generalize effectively to unseen scenarios and for improving the sample efficiency of such policies.

Deep Learning-based Reduced Order Models (DL-ROMs) provide nowadays a well-established class of accurate surrogate models for complex physical systems described by parametrized PDEs, by nonlinearly compressing the solution manifold into a handful of latent coordinates. Until now, design and application of DL-ROMs mainly focused on physically parameterized problems. Within this work, we provide a novel extension of these architectures to problems featuring geometrical variability and parametrized domains, namely, we propose Continuous Geometry-Aware DL-ROMs (CGA-DL-ROMs). In particular, the space-continuous nature of the proposed architecture matches the need to deal with multi-resolution datasets, which are quite common in the case of geometrically parametrized problems. Moreover, CGA-DL-ROMs are endowed with a strong inductive bias that makes them aware of geometrical parametrizations, thus enhancing both the compression capability and the overall performance of the architecture. Within this work, we justify our findings through a thorough theoretical analysis, and we practically validate our claims by means of a series of numerical tests encompassing physically-and-geometrically parametrized PDEs, ranging from the unsteady Navier-Stokes equations for fluid dynamics to advection-diffusion-reaction equations for mathematical biology.

The coupling of Proper Orthogonal Decomposition (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process -- that is, by making them physics-informed -- to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem's solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection-diffusion-reaction equations, to nonlinear problems like the Navier-Stokes equations for fluid flows.

Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution of parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency. This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs in the presence of geometrical variability. In particular, we propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme where a GNN architecture is used to efficiently evolve the system. With respect to the majority of surrogate models, the proposed approach stands out for its ability of tackling problems with parameter dependent spatial domains, while simultaneously generalizing to different geometries and mesh resolutions. We assess the effectiveness of the proposed approach through a series of numerical experiments, involving both two- and three-dimensional problems, showing that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios. We also assess, from a numerical standpoint, the importance of using GNNs, rather than classical dense deep neural networks, for the proposed framework.

However, the solution of parametrized, time-dependent systems of partial differential equations (PDEs) by means of full order models (FOMs) - such as the finite element method - may clash with time and computational budget restrictions. Moreover, using FOMs to explore different scenarios with varying initial conditions and parameter combinations might be a computationally prohibitive task, or even infeasible in several practical applications. Differently from the problem of estimating output quantities of interest that depend on the solution of the differential problem, the computation of the whole solution field is intrinsically high-dimensional, with additional difficulties related to the nonlinear and time-dependent nature of the problem. All these reasons drive the search of efficient, but accurate, reduced order models (ROMs). Among these, the reduced basis method [56, 33, 4] is a very well-known approach, exploiting, e.g., proper orthogonal decomposition (POD) to build a reduced space, either global or local [1, 55], to approximate the solution of the problem. However, despite their accuracy and mathematical, these techniques are in general intrusive [25]. Among machine and deep learning techniques widely used to build surrogate models or emulators to the solution of parametrized, nonlinear, time-dependent system of PDEs, autoencoder (AE) neural networks [27] have recently become a popular strategy because they allow to non-intrusively reduce dimensionality and unveil latent features directly from data streams, without accessing the FOM operators [26, 43, 48, 37]. Their success is due to the expressiveness capacity of neural networks [11, 44, 34], which enables outstanding performances in nonlinear compression and great flexibility in identifying coordinate transformations [46].

Starting from fullorder representations, we apply deep learning techniques to generate accurate, efficient and real-time reduced order models to be used as virtual twin for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches and gyroscopes, also displaying intricate dynamical evolutions like internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows extracting the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems.