A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

Within the framework of parameter dependent PDEs, we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. In order to fully exploit the approximation capabilities of neural networks, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that this minimal dimension is intimately related to the topological properties of the solution manifold, and we provide some theoretical results with particular emphasis on second order elliptic PDEs. Finally, we report numerical experiments where we compare the proposed approach with classical POD-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients

On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields

Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs

Approximation bounds for convolutional neural networks in operator learning

Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature still lack a rigorous justification on their mathematical foundations. Motivated by this fact, in this paper we derive rigorous error bounds for the approximation of nonlinear operators by means of CNN models. More precisely, we address the case in which an operator maps a finite dimensional input $\boldsymbol{\mu}\in\mathbb{R}^{p}$ onto a functional output $u_{\boldsymbol{\mu}}:[0,1]^{d}\to\mathbb{R}$, and a neural network model is used to approximate a discretized version of the input-to-output map. The resulting error estimates provide a clear interpretation of the hyperparameters defining the neural network architecture. All the proofs are constructive, and they ultimately reveal a deep connection between CNNs and the Fourier transform. Finally, we complement the derived error bounds by numerical experiments that illustrate their application

Nonlinear model order reduction for problems with microstructure using mesh informed neural networks

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment

Cleavage of Bcl-2 in oxidant- and cisplatin-induced apoptosis of human melanoma cells

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Mapping Drug Interactions at the Covalent Topoisomerase II-DNA Complex by Bisantrene/Amsacrine Congeners *

To identify structural determinants for the sequence-specific recognition of covalent topoisomerase II-DNA complexes by anti-cancer drugs, we investigated a number of bisantrene congeners, including a 10-azabioisoster, bearing one or two 4, 5-dihydro-1H-imidazol-2-yl hydrazone side chains at positions 1, 4, or 9 of the anthracene ring system. The studied bisantrene/amsacrine (m-AMSA) hybrid and bisantrene isomers were able to poison DNA topoisomerase II with an intermediate activity between those of bisantrene and m-AMSA. Moving the side chain from the central to a lateral ring (from C-9 to C-1/C-4) only slightly modified the drug DNA affinity, whereas it dramatically affected local base preferences of poison-stimulated DNA cleavage. In contrast, switching the planar aromatic systems of bisantrene and m-AMSA did not substantially alter the sequence specificity of drug action. A computer-assisted steric and electrostatic alignment analysis of the test compounds was in agreement with the experimental data, since a common pharmacophore was shared by bisantrene, m-AMSA, and 9-substituted analogs, whereas the 1-substituted isomer showed a radically changed pharmacophoric structure. Thus, the relative space occupancy and electron distribution of putative DNA binding (aromatic rings) and enzyme binding (side chains) moieties are fundamental in directing the specific action of topoisomerase II poisons and in determining the poison pharmacophore

Modulation of cell growth and cisplatin sensitivity by membrane gamma-glutamyltransferase in melanoma cells.

The plasma membrane enzyme c-glutamyltransferase (GGT) is regarded as critical for the maintenance of intracellular levels of glutathione (GSH). GGT expression has been implicated in drug resistance through elevation of intracellular GSH. The dependence of intracellular GSH on GGT expression was not conclusively ascertained. The present study was designed to investigate the role of GGT and of intracellular GSH levels in modulating proliferation and sensitivity to cisplatin of melanoma cells. GGT transfection resulted in increased growth, both in vitro and in tumour xenografts. In addition, GGT-transfected cells exhibited reduced sensitivity to cisplatin associated with lower DNA platination. A decrease in intracellular GSH levels, rather than an increase, was observed in GGT-transfected cells; moreover, in cysteine-deficient conditions, the expression of GGT did not provide transfected cells with the ability of utilising extracellular GSH. In conclusion, these results indicate that GGTactivity confers a growth advantage unrelated with intracellular glutathione supply, and are consistent with the interpretation that cisplatin resistance is the consequence of modifications of cellular pharmacokinetics as a result of extracellular drug inactivation by thiol metabolites originated by GGT-mediated GSH cleavage

Learning high-order interactions for polygenic risk prediction

Within the framework of precision medicine, the stratification of individual genetic susceptibility based on inherited DNA variation has paramount relevance. However, one of the most relevant pitfalls of traditional Polygenic Risk Scores (PRS) approaches is their inability to model complex high-order non-linear SNP-SNP interactions and their effect on the phenotype (e.g. epistasis). Indeed, they incur in a computational challenge as the number of possible interactions grows exponentially with the number of SNPs considered, affecting the statistical reliability of the model parameters as well. In this work, we address this issue by proposing a novel PRS approach, called High-order Interactions-aware Polygenic Risk Score (hiPRS), that incorporates high-order interactions in modeling polygenic risk. The latter combines an interaction search routine based on frequent itemsets mining and a novel interaction selection algorithm based on Mutual Information, to construct a simple and interpretable weighted model of user-specified dimensionality that can predict a given binary phenotype. Compared to traditional PRSs methods, hiPRS does not rely on GWAS summary statistics nor any external information. Moreover, hiPRS differs from Machine Learning-based approaches that can include complex interactions in that it provides a readable and interpretable model and it is able to control overfitting, even on small samples. In the present work we demonstrate through a comprehensive simulation study the superior performance of hiPRS w.r.t. state of the art methods, both in terms of scoring performance and interpretability of the resulting model. We also test hiPRS against small sample size, class imbalance and the presence of noise, showcasing its robustness to extreme experimental settings. Finally, we apply hiPRS to a case study on real data from DACHS cohort, defining an interaction-aware scoring model to predict mortality of stage II-III Colon-Rectal Cancer patients treated with oxaliplatin