Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

In recent years, many accurate decision support systems have been constructed as black boxes, that is as systems that hide their internal logic to the user. This lack of explanation constitutes both a practical and an ethical issue. The literature reports many approaches aimed at overcoming this crucial weakness, sometimes at the cost of sacrificing accuracy for interpretability. The applications in which black box decision systems can be used are various, and each approach is typically developed to provide a solution for a specific problem and, as a consequence, it explicitly or implicitly delineates its own definition of interpretability and explanation. The aim of this article is to provide a classification of the main problems addressed in the literature with respect to the notion of explanation and the type of black box system. Given a problem definition, a black box type, and a desired explanation, this survey should help the researcher to find the proposals more useful for his own work. The proposed classification of approaches to open black box models should also be useful for putting the many research open questions in perspective.

https://dl.acm.org/doi/pdf/10.1145/3236009

A Survey of Methods for Explaining Black Box Models

In the last years, Artificial Intelligence (AI) has achieved a notable momentum that may deliver the best of expectations over many application sectors across the field. For this to occur, the entire community stands in front of the barrier of explainability, an inherent problem of AI techniques brought by sub-symbolism (e.g. ensembles or Deep Neural Networks) that were not present in the last hype of AI. Paradigms underlying this problem fall within the so-called eXplainable AI (XAI) field, which is acknowledged as a crucial feature for the practical deployment of AI models. This overview examines the existing literature in the field of XAI, including a prospect toward what is yet to be reached. We summarize previous efforts to define explainability in Machine Learning, establishing a novel definition that covers prior conceptual propositions with a major focus on the audience for which explainability is sought. We then propose and discuss about a taxonomy of recent contributions related to the explainability of different Machine Learning models, including those aimed at Deep Learning methods for which a second taxonomy is built. This literature analysis serves as the background for a series of challenges faced by XAI, such as the crossroads between data fusion and explainability. Our prospects lead toward the concept of Responsible Artificial Intelligence, namely, a methodology for the large-scale implementation of AI methods in real organizations with fairness, model explainability and accountability at its core. Our ultimate goal is to provide newcomers to XAI with a reference material in order to stimulate future research advances, but also to encourage experts and professionals from other disciplines to embrace the benefits of AI in their activity sectors, without any prior bias for its lack of interpretability.

Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI.

Machine learning (ML) encompasses a broad range of algorithms and modeling tools used for a vast array of data processing tasks, which has entered most scientific disciplines in recent years. This article reviews in a selective way the recent research on the interface between machine learning and the physical sciences. This includes conceptual developments in ML motivated by physical insights, applications of machine learning techniques to several domains in physics, and cross fertilization between the two fields. After giving a basic notion of machine learning methods and principles, examples are described of how statistical physics is used to understand methods in ML. This review then describes applications of ML methods in particle physics and cosmology, quantum many-body physics, quantum computing, and chemical and material physics. Research and development into novel computing architectures aimed at accelerating ML are also highlighted. Each of the sections describe recent successes as well as domain-specific methodology and challenges.

Machine learning and the physical sciences

With rapid progress and significant successes in a wide spectrum of applications, deep learning is being applied in many safety-critical environments. However, deep neural networks (DNNs) have been recently found vulnerable to well-designed input samples called adversarial examples . Adversarial perturbations are imperceptible to human but can easily fool DNNs in the testing/deploying stage. The vulnerability to adversarial examples becomes one of the major risks for applying DNNs in safety-critical environments. Therefore, attacks and defenses on adversarial examples draw great attention. In this paper, we review recent findings on adversarial examples for DNNs, summarize the methods for generating adversarial examples, and propose a taxonomy of these methods. Under the taxonomy, applications for adversarial examples are investigated. We further elaborate on countermeasures for adversarial examples. In addition, three major challenges in adversarial examples and the potential solutions are discussed.

Adversarial Examples: Attacks and Defenses for Deep Learning

Despite their great success, there is still no comprehensive theoretical understanding of learning with Deep Neural Networks (DNNs) or their inner organization. Previous work proposed to analyze DNNs in the \textit{Information Plane}; i.e., the plane of the Mutual Information values that each layer preserves on the input and output variables. They suggested that the goal of the network is to optimize the Information Bottleneck (IB) tradeoff between compression and prediction, successively, for each layer. In this work we follow up on this idea and demonstrate the effectiveness of the Information-Plane visualization of DNNs. Our main results are: (i) most of the training epochs in standard DL are spent on {\emph compression} of the input to efficient representation and not on fitting the training labels. (ii) The representation compression phase begins when the training errors becomes small and the Stochastic Gradient Decent (SGD) epochs change from a fast drift to smaller training error into a stochastic relaxation, or random diffusion, constrained by the training error value. (iii) The converged layers lie on or very close to the Information Bottleneck (IB) theoretical bound, and the maps from the input to any hidden layer and from this hidden layer to the output satisfy the IB self-consistent equations. This generalization through noise mechanism is unique to Deep Neural Networks and absent in one layer networks. (iv) The training time is dramatically reduced when adding more hidden layers. Thus the main advantage of the hidden layers is computational. This can be explained by the reduced relaxation time, as this it scales super-linearly (exponentially for simple diffusion) with the information compression from the previous layer.

Opening the Black Box of Deep Neural Networks via Information

/pdf/tabular-data-deep-learning-is-not-all-you-need-h5yezwdy.pdf

Tabular data: Deep learning is not all you need

Associative learning of pure tones is known to cause tonotopic map expansion in the auditory cortex (ACx), but the function this plasticity sub-serves is unclear. We developed an automated training platform called the "Educage," which was used to train mice on a go/no-go auditory discrimination task to their perceptual limits, for difficult discriminations among pure tones or natural sounds. Spiking responses of excitatory and inhibitory parvalbumin (PV+) L2/3 neurons in mouse ACx revealed learning-induced overrepresentation of the learned frequencies, as expected from previous literature. The coordinated plasticity of excitatory and inhibitory neurons supports a role for PV+ neurons in homeostatic maintenance of excitation-inhibition balance within the circuit. Using a novel computational model to study auditory tuning curves, we show that overrepresentation of the learned tones does not necessarily improve discrimination performance of the network to these tones. In a separate set of experiments, we trained mice to discriminate among natural sounds. Perceptual learning of natural sounds induced "sparsening" and decorrelation of the neural response, consequently improving discrimination of these complex sounds. This signature of plasticity in A1 highlights its role in coding natural sounds.

Neural Correlates of Learning Pure Tones or Natural Sounds in the Auditory Cortex.

In this preliminary work, we study the generalization properties of infinite ensembles of infinitely-wide neural networks. Amazingly, this model family admits tractable calculations for many information-theoretic quantities. We report analytical and empirical investigations in the search for signals that correlate with generalization.

Ravid Shwartz-Ziv

Papers

Opening the Black Box of Deep Neural Networks via Information

Tabular data: Deep learning is not all you need

Tabular data: Deep learning is not all you need

Neural Correlates of Learning Pure Tones or Natural Sounds in the Auditory Cortex.

Information in Infinite Ensembles of Infinitely-Wide Neural Networks