We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.
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ACM/IEEE第23屆模型驅動工程語言和系統國際會議，是模型驅動軟件和系統工程的首要會議系列，由ACM-SIGSOFT和IEEE-TCSE支持組織。自1998年以來，模型涵蓋了建模的各個方面，從語言和方法到工具和應用程序。模特的參加者來自不同的背景，包括研究人員、學者、工程師和工業專業人士。MODELS 2019是一個論壇，參與者可以圍繞建模和模型驅動的軟件和系統交流前沿研究成果和創新實踐經驗。今年的版本將為建模社區提供進一步推進建模基礎的機會，并在網絡物理系統、嵌入式系統、社會技術系統、云計算、大數據、機器學習、安全、開源等新興領域提出建模的創新應用以及可持續性。
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In this paper, we introduce a new representation for team-coordinated game-theoretic decision making, which we coin team belief DAG form. In our representation, at every timestep, a team coordinator observes the information that is public to all its members, and then decides on a prescription for all the possible states consistent with its observations. Our representation unifies and extends recent approaches to team coordination. Similar to the approach of Carminati et al (2021), our team belief DAG form can be used to capture adversarial team games, and enables standard, out-of-the-box game-theoretic techniques including no-regret learning (e.g., CFR and its state-of-the-art modern variants such as DCFR and PCFR+) and first-order methods. However, our representation can be exponentially smaller, and can be viewed as a lossless abstraction of theirs into a directed acyclic graph. In particular, like the LP-based algorithm of Zhang & Sandholm (2022), the size of our representation scales with the amount of information uncommon to the team; in fact, using linear programming on top of our team belief DAG form to solve for a team correlated equilibrium in an adversarial team games recovers almost exactly their algorithm. Unlike that paper, however, our representation explicitly exposes the structure of the decision space, which is what enables the aforementioned game-theoretic techniques.
We address a three-tier data-driven approach to solve the inverse problem in complex systems modelling from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps using leave-one-out cross-validation (LOOCV) to both identify the intrinsic dimension of the manifold where the emergent dynamics evolve and for feature selection over the parametric space. In the second step, based on the selected features, we learn the right-hand-side of the effective partial differential equations (PDEs) using two machine learning schemes, namely shallow Feedforward Neural Networks (FNNs) with two hidden layers and single-layer Random Projection Networks(RPNNs) which basis functions are constructed using an appropriate random sampling approach. Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting the numerical bifurcation analysis toolkit. For our illustrations, we implemented the proposed method to construct the one-parameter bifurcation diagram of the 1D FitzHugh-Nagumo PDEs from data generated by $D1Q3$ Lattice Boltzmann simulations. The proposed method was quite effective in terms of numerical accuracy regarding the construction of the coarse-scale bifurcation diagram. Furthermore, the proposed RPNN scheme was $\sim$ 20 to 30 times less costly regarding the training phase than the traditional shallow FNNs, thus arising as a promising alternative to deep learning for solving the inverse problem for high-dimensional PDEs.
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. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by 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.
With the field of rigid-body robotics having matured in the last fifty years, routing, planning, and manipulation of deformable objects have emerged in recent years as a more untouched research area in many fields ranging from surgical robotics to industrial assembly and construction. Routing approaches for deformable objects which rely on learned implicit spatial representations (e.g., Learning-from-Demonstration methods) make them vulnerable to changes in the environment and the specific setup. On the other hand, algorithms that entirely separate the spatial representation of the deformable object from the routing and manipulation, often using a representation approach independent of planning, result in slow planning in high dimensional space. This paper proposes a novel approach to spatial representation combined with route planning that allows efficient routing of deformable one-dimensional objects (e.g., wires, cables, ropes, threads). The spatial representation is based on the geometrical decomposition of the space into convex subspaces, which allows an efficient coding of the configuration. Having such a configuration, the routing problem can be solved using a dynamic programming matching method with a quadratic time and space complexity. The proposed method couples the routing and efficient configuration for improved planning time. Our tests and experiments show the method correctly computing the next manipulation action in sub-millisecond time and accomplishing various routing and manipulation tasks.
Integrated and Adaptive Guidance and Control for Endoatmospheric Missiles via Reinforcement Learning
We apply a reinforcement meta-learning framework to optimize an integrated and adaptive guidance and flight control system for an air-to-air missile. The system is implemented as a policy that maps navigation system outputs directly to commanded rates of change for the missile's control surface deflections. The system induces intercept trajectories against a maneuvering target that satisfy control constraints on fin deflection angles, and path constraints on look angle and load. We test the optimized system in a six degrees-of-freedom simulator that includes a non-linear radome model and a strapdown seeker model, and demonstrate that the system adapts to both a large flight envelope and off-nominal flight conditions including perturbation of aerodynamic coefficient parameters and center of pressure locations, and flexible body dynamics. Moreover, we find that the system is robust to the parasitic attitude loop induced by radome refraction and imperfect seeker stabilization. We compare our system's performance to a longitudinal model of proportional navigation coupled with a three loop autopilot, and find that our system outperforms this benchmark by a large margin. Additional experiments investigate the impact of removing the recurrent layer from the policy and value function networks, performance with an infrared seeker, and flexible body dynamics.
The problem of nonprehensile manipulation of a stick in three-dimensional space using intermittent impulsive forces is considered. The objective is to juggle the stick between a sequence of configurations that are rotationally symmetric about the vertical axis. The dynamics of the stick is described by five generalized coordinates and three control inputs. Between two consecutive rotationally symmetric configurations, the dynamics is conveniently represented by a Poincar\'e map in the reference frame of the juggler. Stabilization of the orbit associated with a desired juggling motion is accomplished by stabilizing a fixed point on the Poincar\'e map. The Impulse Controlled Poincar\'e Map approach is used to stabilize the orbit, and numerical simulations are used demonstrate convergence to the desired juggling motion from an arbitrary initial configuration. In the limiting case, where consecutive rotationally symmetric configurations are chosen arbitrarily close, it is shown that the dynamics reduces to that of steady precession of the stick on a hoop.
We propose a new framework to reconstruct a stochastic process $\left\{\mathbb{P}_{t}: t \in[0, T]\right\}$ using only samples from its marginal distributions, observed at start and end times $0$ and $T$. This reconstruction is useful to infer population dynamics, a crucial challenge, e.g., when modeling the time-evolution of cell populations from single-cell sequencing data. Our general framework encompasses the more specific Schr\"odinger bridge (SB) problem, where $\mathbb{P}_{t}$ represents the evolution of a thermodynamic system at almost equilibrium. Estimating such bridges is notoriously difficult, motivating our proposal for a novel adaptive scheme called the GSBflow. Our goal is to rely on Gaussian approximations of the data to provide the reference stochastic process needed to estimate SB. To that end, we solve the \acs{SB} problem with Gaussian marginals, for which we provide, as a central contribution, a closed-form solution and SDE-representation. We use these formulas to define the reference process used to estimate more complex SBs, and show that this does indeed help with its numerical solution. We obtain notable improvements when reconstructing both synthetic processes and single-cell genomics experiments.
We investigate the complexity of explicit construction problems, where the goal is to produce a particular object of size $n$ possessing some pseudorandom property in time polynomial in $n$. We give overwhelming evidence that $\bf{APEPP}$, defined originally by Kleinberg et al., is the natural complexity class associated with explicit constructions of objects whose existence follows from the probabilistic method, by placing a variety of such construction problems in this class. We then demonstrate that a result of Je\v{r}\'{a}bek on provability in Bounded Arithmetic, when reinterpreted as a reduction between search problems, shows that constructing a truth table of high circuit complexity is complete for $\bf{APEPP}$ under $\bf{P}^{\bf{NP}}$ reductions. This illustrates that Shannon's classical proof of the existence of hard boolean functions is in fact a $\textit{universal}$ probabilistic existence argument: derandomizing his proof implies a generic derandomization of the probabilistic method. As a corollary, we prove that $\bf{EXP}^{\bf{NP}}$ contains a language of circuit complexity $2^{n^{\Omega(1)}}$ if and only if it contains a language of circuit complexity $\frac{2^n}{2n}$. Finally, for several of the problems shown to lie in $\bf{APEPP}$, we demonstrate direct polynomial time reductions to the explicit construction of hard truth tables.
Promoting behavioural diversity is critical for solving games with non-transitive dynamics where strategic cycles exist, and there is no consistent winner (e.g., Rock-Paper-Scissors). Yet, there is a lack of rigorous treatment for defining diversity and constructing diversity-aware learning dynamics. In this work, we offer a geometric interpretation of behavioural diversity in games and introduce a novel diversity metric based on \emph{determinantal point processes} (DPP). By incorporating the diversity metric into best-response dynamics, we develop \emph{diverse fictitious play} and \emph{diverse policy-space response oracle} for solving normal-form games and open-ended games. We prove the uniqueness of the diverse best response and the convergence of our algorithms on two-player games. Importantly, we show that maximising the DPP-based diversity metric guarantees to enlarge the \emph{gamescape} -- convex polytopes spanned by agents' mixtures of strategies. To validate our diversity-aware solvers, we test on tens of games that show strong non-transitivity. Results suggest that our methods achieve much lower exploitability than state-of-the-art solvers by finding effective and diverse strategies.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena. We train a Restricted Boltzmann Machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods. From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system. We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near criticality region of the Ising model. In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the Renormalization Group (RG) flow of the lattice model. Our results suggest an alternative explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points. Then from the reconstructed data we deduce the critical exponent associated to the magnetization to find satisfactory agreement with the actual physical value. We assume no prior knowledge about the criticality of the system and its Hamiltonian.
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