Sylvie Putot

Franck Djeumou, Cyrus Neary, Eric Goubault et al.

Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every forward evaluation of a NODE requires numerical integration of the neural network used to capture the system dynamics, making their training prohibitively expensive. Existing works rely on off-the-shelf adaptive step-size numerical integration schemes, which often require an excessive number of evaluations of the underlying dynamics network to obtain sufficient accuracy for training. By contrast, we accelerate the evaluation and the training of NODEs by proposing a data-driven approach to their numerical integration. The proposed Taylor-Lagrange NODEs (TL-NODEs) use a fixed-order Taylor expansion for numerical integration, while also learning to estimate the expansion's approximation error. As a result, the proposed approach achieves the same accuracy as adaptive step-size schemes while employing only low-order Taylor expansions, thus greatly reducing the computational cost necessary to integrate the NODE. A suite of numerical experiments, including modeling dynamical systems, image classification, and density estimation, demonstrate that TL-NODEs can be trained more than an order of magnitude faster than state-of-the-art approaches, without any loss in performance.

Franck Djeumou, Cyrus Neary, Eric Goubault et al.

Effective inclusion of physics-based knowledge into deep neural network models of dynamical systems can greatly improve data efficiency and generalization. Such a-priori knowledge might arise from physical principles (e.g., conservation laws) or from the system's design (e.g., the Jacobian matrix of a robot), even if large portions of the system dynamics remain unknown. We develop a framework to learn dynamics models from trajectory data while incorporating a-priori system knowledge as inductive bias. More specifically, the proposed framework uses physics-based side information to inform the structure of the neural network itself, and to place constraints on the values of the outputs and the internal states of the model. It represents the system's vector field as a composition of known and unknown functions, the latter of which are parametrized by neural networks. The physics-informed constraints are enforced via the augmented Lagrangian method during the model's training. We experimentally demonstrate the benefits of the proposed approach on a variety of dynamical systems -- including a benchmark suite of robotics environments featuring large state spaces, non-linear dynamics, external forces, contact forces, and control inputs. By exploiting a-priori system knowledge during training, the proposed approach learns to predict the system dynamics two orders of magnitude more accurately than a baseline approach that does not include prior knowledge, given the same training dataset.

Maria Luiza Costa Vianna, Eric Goubault, Sylvie Putot

We study learning based controllers as a replacement for model predictive controllers (MPC) for the control of autonomous vehicles. We concentrate for the experiments on the simple yet representative bicycle model. We compare training by supervised learning and by reinforcement learning. We also discuss the neural net architectures so as to obtain small nets with the best performances. This work aims at producing controllers that can both be embedded on real-time platforms and amenable to verification by formal methods techniques.

Eric Goubault, Sébastien Palumby, Sylvie Putot et al.

This paper studies the problem of range analysis for feedforward neural networks, which is a basic primitive for applications such as robustness of neural networks, compliance to specifications and reachability analysis of neural-network feedback systems. Our approach focuses on ReLU (rectified linear unit) feedforward neural nets that present specific difficulties: approaches that exploit derivatives do not apply in general, the number of patterns of neuron activations can be quite large even for small networks, and convex approximations are generally too coarse. In this paper, we employ set-based methods and abstract interpretation that have been very successful in coping with similar difficulties in classical program verification. We present an approach that abstracts ReLU feedforward neural networks using tropical polyhedra. We show that tropical polyhedra can efficiently abstract ReLU activation function, while being able to control the loss of precision due to linear computations. We show how the connection between ReLU networks and tropical rational functions can provide approaches for range analysis of ReLU neural networks.

Nicola Bernini, Mikhail Bessa, Rémi Delmas et al.

We explore the reinforcement learning approach to designing controllers by extensively discussing the case of a quadcopter attitude controller. We provide all details allowing to reproduce our approach, starting with a model of the dynamics of a crazyflie 2.0 under various nominal and non-nominal conditions, including partial motor failures and wind gusts. We develop a robust form of a signal temporal logic to quantitatively evaluate the vehicle's behavior and measure the performance of controllers. The paper thoroughly describes the choices in training algorithms, neural net architecture, hyperparameters, observation space in view of the different performance metrics we have introduced. We discuss the robustness of the obtained controllers, both to partial loss of power for one rotor and to wind gusts and finish by drawing conclusions on practical controller design by reinforcement learning.

Benjamin Martin, Khalil Ghorbal, Eric Goubault et al.

We formally verify a hybrid control law designed to perform a station keeping maneuver for a planar vehicle. Such maneuver requires that the vehicle reaches a neighborhood of its station in finite time and remains in it while waiting for further instructions. We model the dynamics as well as the control law as a hybrid program and formally verify both the reachability and safety properties involved. We highlight in particular the automated generation of invariant regions which turns out to be crucial in performing such verification. We use the theorem prover Keymaera X to discharge some of the generated proof obligations.

Eric Goubault, Sylvie Putot

A desirable property of control systems is to be robust to inputs, that is small perturbations of the inputs of a system will cause only small perturbations on its outputs. But it is not clear whether this property is maintained at the implementation level, when two close inputs can lead to very different execution paths. The problem becomes particularly crucial when considering finite precision implementations, where any elementary computation can be affected by a small error. In this context, almost every test is potentially unstable, that is, for a given input, the computed (finite precision) path may differ from the ideal (same computation in real numbers) path. Still, state-of-the-art error analyses do not consider this possibility and rely on the stable test hypothesis, that control flows are identical. If there is a discontinuity between the treatments in the two branches, that is the conditional block is not robust to uncertainties, the error bounds can be unsound. We propose here a new abstract-interpretation based error analysis of finite precision implementations, which is sound in presence of unstable tests. It automatically bounds the discontinuity error coming from the difference between the float and real values when there is a path divergence, and introduces a new error term labeled by the test that introduced this potential discontinuity. This gives a tractable error analysis, implemented in our static analyzer FLUCTUAT: we present results on representative extracts of control programs.

Khalil Ghorbal, Eric Goubault, Sylvie Putot

We define and study a new abstract domain which is a fine-grained combination of zonotopes with polyhedric domains such as the interval, octagon, linear templates or polyhedron domain. While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e. intersections) efficiently. This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid sys- tems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic over-approximation of the intersection. We describe some examples and an im- plementation of our method in the APRON library, and discuss some further in- teresting combinations of zonotopes with non-linear or non-convex domains such as quadratic templates and maxplus polyhedra.

Eric Goubault, Sylvie Putot

We completely describe a new domain for abstract interpretation of numerical programs. Fixpoint iteration in this domain is proved to converge to finite precise invariants for (at least) the class of stable linear recursive filters of any order. Good evidence shows it behaves well also for some non-linear schemes. The result, and the structure of the domain, rely on an interesting interplay between order and topology.