M. Ortíz

Agustin Bompadre, Bernd Schmidt, Michael Ortiz

This work is concerned with the formulation of a general framework for the analysis of meshfree approximation schemes and with the convergence analysis of the Local Maximum-Entropy (LME) scheme as a particular example. We provide conditions for the convergence in Sobolev spaces of schemes that are n-consistent, in the sense of exactly reproducing polynomials of degree less or equal to n, and whose basis functions are of rapid decay. The convergence of the LME in a locally Sobolev space follows as a direct application of the general theory. The analysis shows that the convergence order is linear in h, a measure of the density of the point set. The analysis also shows how to parameterize the LME scheme for optimal convergence. Because of the convex approximation property of LME, its behavior near the boundary is singular and requires additional analysis. For the particular case of polyhedral domains we show that, away from a small singular part of the boundary, any Sobolev function can be approximated by means of the LME scheme. With the aid of a capacity argument, we further obtain approximation results with truncated LME basis functions in H1 and for spatial dimension d > 2.

Eky Febrianto, Yiren Wang, Burigede Liu et al.

Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.

Livio Fedeli, Anna Pandolfi, Michael Ortiz

We develop an Optimal Transportation Meshfree (OTM) particle method for advection-diffusion in which the concentration or density of the diffusive species is approximated by Dirac measures. We resort to an incremental variational principle for purposes of time discretization of the diffusive step. This principle characterizes the evolution of the density as a competition between the Wasserstein distance between two consecutive densities and entropy. Exploiting the structure of the Euler-Lagrange equations, we approximate the density as a collection of Diracs. The interpolation of the incremental transport map is effected through mesh-free max-ent interpolation. Remarkably, the resulting update is geometrically exact with respect to advection and volume. We present three-dimensional examples of application that illustrate the scope and robustness of the method.

Kai Partmann, Christian Wieners, Michael Ortiz et al.

Peridynamics formulates the balance of linear momentum as an integro-differential equation, making it naturally suited for fracture modeling without special treatment of discontinuities. The bond-associated correspondence formulation provides a highly accurate peridynamic framework by computing bond-wise deformation gradients that are free of zero-energy modes and yield accurate results even near boundaries. However, the traditional fracture approach based on irreversible bond deletion can compromise this formulation, as the progressive removal of bonds degrades the nonlocal approximation of the deformation gradient and can lead to numerical instabilities. In this work, a novel phase-field peridynamics approach is introduced that avoids these instabilities. Instead of deleting bonds, the energetic contribution of each bond is continuously degraded through a bond phase-field parameter, while a separate kinematic degradation function preserves the accuracy of the nonlocal deformation gradient approximation. The normalization constant ensuring thermodynamic consistency with Griffith's fracture theory is derived analytically for general spherical kernel functions as a ratio of two one-dimensional integrals. Numerical examples including mode I and mode II fracture, the boundary tension test with different kernel functions and horizon ratios, and the Kalthoff-Winkler experiment demonstrate the stability, accuracy, and consistency of the proposed approach.

Vince Jankovics, Michael Garcia Ortiz, Eduardo Alonso

Recent deep reinforcement learning (DRL) successes rely on end-to-end learning from fixed-size observational inputs (e.g. image, state-variables). However, many challenging and interesting problems in decision making involve observations or intermediary representations which are best described as a set of entities: either the image-based approach would miss small but important details in the observations (e.g. ojects on a radar, vehicles on satellite images, etc.), the number of sensed objects is not fixed (e.g. robotic manipulation), or the problem simply cannot be represented in a meaningful way as an image (e.g. power grid control, or logistics). This type of structured representations is not directly compatible with current DRL architectures, however, there has been an increase in machine learning techniques directly targeting structured information, potentially addressing this issue. We propose to combine recent advances in set representations with slot attention and graph neural networks to process structured data, broadening the range of applications of DRL algorithms. This approach allows to address entity-based problems in an efficient and scalable way. We show that it can improve training time and robustness significantly, and demonstrate their potential to handle structured as well as purely visual domains, on multiple environments from the Atari Learning Environment and Simple Playgrounds.

Kerstin Weinberg, Laurent Strainier, Sergio Conti et al.

We resort to game theory in order to formulate Data-Driven methods for solid mechanics in which stress and strain players pursue different objectives. The objective of the stress player is to minimize the discrepancy to a material data set, whereas the objective of the strain player is to ensure the admissibility of the mechanical state, in the sense of compatibility and equilibrium. We show that, unlike the cooperative Data-Driven games proposed in the past, the new non-cooperative Data-Driven games identify an effective material law from the data and reduce to conventional displacement boundary-value problems, which facilitates their practical implementation. However, unlike supervised machine learning methods, the proposed non-cooperative Data-Driven games are unsupervised, ansatz-free and parameter-free. In particular, the effective material law is learned from the data directly, without recourse to regression to a parameterized class of functions such as neural networks. We present analysis that elucidates sufficient conditions for convergence of the Data-Driven solutions with respect to the data. We also present selected examples of implementation and application that demonstrate the range and versatility of the approach.

Hugo Caselles-Dupré, Michael Garcia-Ortiz, David Filliat

Affordances are the possibilities of actions the environment offers to the individual. Ordinary objects (hammer, knife) usually have many affordances (grasping, pounding, cutting), and detecting these allow artificial agents to understand what are their possibilities in the environment, with obvious application in Robotics. Proposed benchmarks and state-of-the-art prediction models for supervised affordance segmentation are usually modifications of popular object segmentation models such as Mask R-CNN. We observe that theoretically, these popular object segmentation methods should be sufficient for detecting affordances masks. So we ask the question: is it necessary to tailor new architectures to the problem of learning affordances? We show that applying the out-of-the-box Mask R-CNN to the problem of affordances segmentation outperforms the current state-of-the-art. We conclude that the problem of supervised affordance segmentation is included in the problem of object segmentation and argue that better benchmarks for affordance learning should include action capacities.

Hugo Caselles-Dupré, Michael Garcia-Ortiz, David Filliat

We introduce SCOD (Sensory Commutativity Object Detection), an active method for movable and immovable object detection. SCOD exploits the commutative properties of action sequences, in the scenario of an embodied agent equipped with first-person sensors and a continuous motor space with multiple degrees of freedom. SCOD is based on playing an action sequence in two different orders from the same starting point and comparing the two final observations obtained after each sequence. Our experiments on 3D realistic robotic setups (iGibson) demonstrate the accuracy of SCOD and its generalization to unseen environments and objects. We also successfully apply SCOD on a real robot to further illustrate its generalization properties. With SCOD, we aim at providing a novel way of approaching the problem of object discovery in the context of a naive embodied agent. We provide code and a supplementary video.

Alban Laflaquière, Michael Garcia Ortiz

Despite its omnipresence in robotics application, the nature of spatial knowledge and the mechanisms that underlie its emergence in autonomous agents are still poorly understood. Recent theoretical works suggest that the Euclidean structure of space induces invariants in an agent's raw sensorimotor experience. We hypothesize that capturing these invariants is beneficial for sensorimotor prediction and that, under certain exploratory conditions, a motor representation capturing the structure of the external space should emerge as a byproduct of learning to predict future sensory experiences. We propose a simple sensorimotor predictive scheme, apply it to different agents and types of exploration, and evaluate the pertinence of these hypotheses. We show that a naive agent can capture the topology and metric regularity of its sensor's position in an egocentric spatial frame without any a priori knowledge, nor extraneous supervision.

Louis Annabi, Michael Garcia Ortiz

Unsupervised learning of compact and relevant state representations has been proved very useful at solving complex reinforcement learning tasks. In this paper, we propose a recurrent capsule network that learns such representations by trying to predict the future observations in an agent's trajectory.

Hugo Caselles-Dupré, Michael Garcia-Ortiz, David Filliat

We consider the problem of building a state representation model in a continual fashion. As the environment changes, the aim is to efficiently compress the sensory state's information without losing past knowledge. The learned features are then fed to a Reinforcement Learning algorithm to learn a policy. We propose to use Variational Auto-Encoders for state representation, and Generative Replay, i.e. the use of generated samples, to maintain past knowledge. We also provide a general and statistically sound method for automatic environment change detection. Our method provides efficient state representation as well as forward transfer, and avoids catastrophic forgetting. The resulting model is capable of incrementally learning information without using past data and with a bounded system size.

Alban Laflaquière, Nikolas Hemion, Michaël Garcia Ortiz et al.

Sensorimotor contingency theory offers a promising account of the nature of perception, a topic rarely addressed in the robotics community. We propose a developmental framework to address the problem of the autonomous acquisition of sensorimotor contingencies by a naive robot. While exploring the world, the robot internally encodes contingencies as predictive models that capture the structure they imply in its sensorimotor experience. Three preliminary applications are presented to illustrate our approach to the acquisition of perceptive abilities: discovering the environment, discovering objects, and discovering a visual field.

Alban Laflaquière, Michael Garcia Ortiz

Despite its omnipresence in robotics application, the nature of spatial knowledge and the mechanisms that underlie its emergence in autonomous agents are still poorly understood. Recent theoretical work suggests that the concept of space can be grounded by capturing invariants induced by the structure of space in an agent's raw sensorimotor experience. Moreover, it is hypothesized that capturing these invariants is beneficial for a naive agent trying to predict its sensorimotor experience. Under certain exploratory conditions, spatial representations should thus emerge as a byproduct of learning to predict. We propose a simple sensorimotor predictive scheme, apply it to different agents and types of exploration, and evaluate the pertinence of this hypothesis. We show that a naive agent can capture the topology and metric regularity of its spatial configuration without any a priori knowledge, nor extraneous supervision.

Michael Garcia Ortiz, Alban Laflaquière

Robots act in their environment through sequences of continuous motor commands. Because of the dimensionality of the motor space, as well as the infinite possible combinations of successive motor commands, agents need compact representations that capture the structure of the resulting displacements. In the case of an autonomous agent with no a priori knowledge about its sensorimotor apparatus, this compression has to be learned. We propose to use Recurrent Neural Networks to encode motor sequences into a compact representation, which is used to predict the consequence of motor sequences in term of sensory changes. We show that sensory prediction can successfully guide the compression of motor sequences into representations that are organized topologically in term of spatial displacement.

Thibaut Kulak, Michael Garcia Ortiz

In order to explore and act autonomously in an environment, an agent needs to learn from the sensorimotor information that is captured while acting. By extracting the regularities in this sensorimotor stream, it can learn a model of the world, which in turn can be used as a basis for action and exploration. This requires the acquisition of compact representations from a possibly high dimensional raw observation, which is noisy and ambiguous. In this paper, we learn sensory representations from sensorimotor prediction. We propose a model which integrates sensorimotor information over time, and project it in a sensory representation which is useful for prediction. We emphasize on a simple example the role of motor and memory for learning sensory representations.