Publications by Eduardo D. Sontag in year 2024 |
Articles in journal or book chapters |
This paper studies the effect of perturbations on the gradient flow of a general constrained nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectory of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. Interestingly, we show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow. |
Deep neural network autoencoders are routinely used computationally for model reduction. They allow recognizing the intrinsic dimension of data that lie in a k-dimensional subset K of an input Euclidean space $\R^n$. The underlying idea is to obtain both an encoding layer that maps $\R^n$ into $\R^k$ (called the bottleneck layer or the space of latent variables) and a decoding layer that maps $\R^k$ back into $\R^n$, in such a way that the input data from the set K is recovered when composing the two maps. This is achieved by adjusting parameters (weights) in the network to minimize the discrepancy between the input and the reconstructed output. Since neural networks (with continuous activation functions) compute continuous maps, the existence of a network that achieves perfect reconstruction would imply that K is homeomorphic to a k-dimensional subset of $\R^k$, so clearly there are topological obstructions to finding such a network. On the other hand, in practice the technique is found to "work" well, which leads one to ask if there is a way to explain this effectiveness. We show that, up to small errors, indeed the method is guaranteed to work. This is done by appealing to certain facts from differential geometry. A computational example is also included to illustrate the ideas. |
This paper defines antifragility for dynamical systems as convexity of a newly introduced "logarithmic rate" of dynamical systems. It shows how to compute this rate for positive linear systems, and it interprets antifragility in terms of pulsed alternations of extreme strategies in comparison to average uniform strategies. |
Single-cell omics technologies can measure millions of cells for up to thousands of biomolecular features, which enables the data-driven study of highly complex biological networks. However, these high-throughput experimental techniques often cannot track individual cells over time, thus complicating the understanding of dynamics such as the time trajectories of cell states. These ``dynamical phenotypes'' are key to understanding biological phenomena such as differentiation fates. We show by mathematical analysis that, in spite of high-dimensionality and lack of individual cell traces, three timepoints of single-cell omics data are theoretically necessary and sufficient in order to uniquely determine the network interaction matrix and associated dynamics. Moreover, we show through numerical simulations that an interaction matrix can be accurately determined with three or more timepoints even in the presence of sampling and measurement noise typical of single-cell omics. Our results can guide the design of single-cell omics time-course experiments, and provide a tool for data-driven phase-space analysis. |
Differentiation within multicellular organisms is a complex process that helps to establish spatial patterning and tissue formation within the body. Often, the differentiation of cells is governed by morphogens and intercellular signaling molecules that guide the fate of each cell, frequently using toggle-like regulatory components. Synthetic biologists have long sought to recapitulate patterned differentiation with engineered cellular communities, and various methods for differentiating bacteria have been invented. Here, we couple a synthetic corepressive toggle switch with intercellular signaling pathways to create a “quorum-sensing toggle”. We show that this circuit not only exhibits population-wide bistability in a well-mixed liquid environment but also generates patterns of differentiation in colonies grown on agar containing an externally supplied morphogen. If coupled to other metabolic processes, circuits such as the one described here would allow for the engineering of spatially patterned, differentiated bacteria for use in biomaterials and bioelectronics. |
We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the impact of the attack over the bilinear dynamical network. In particular, the H2-norm of bilinear systems is known to capture robustness and performance properties analogous to its linear counterpart and provides valuable insights for identifying which edges arem ost sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, and brute-force approaches show poor scalability. However, we show that the H2-norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulation. |
Conference articles |
In the context of epigenetic transformations in cancer metastasis, a puzzling effect was recently discovered, in which the elimination (knock-out) of an activating regulatory element leads to increased (rather than decreased) activity of the element being regulated. It has been postulated that this paradoxical behavior can be explained by activating and repressing transcription factors competing for binding to other possible targets. It is very difficult to prove this hypothesis in mammalian cells, due to the large number of potential players and the complexity of endogenous intracellular regulatory networks. Instead, this paper analyzes this issue through an analogous synthetic biology construct which aims to reproduce the paradoxical behavior using standard bacterial gene expression networks. The paper first reviews the motivating cancer biology work, and then describes a proposed synthetic construct. A mathematical model is formulated, and basic properties of uniqueness of steady states and convergence to equilibria are established, as well as an identification of parameter regimes which should lead to observing such paradoxical phenomena (more activator leads to less activity at steady state). A proof is also given to show that this is a steady-state property, and for initial transients the phenomenon will not be observed. This work adds to the general line of work of resource competition in synthetic circuits. |
In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $\omega$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $\omega$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $\omega$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are contractive with respect to any strictly convex norm (such as $\ell^p$ for any $p ot=1,\infty$), and if there is at least one bounded solution, then the $\omega$-limit set of every trajectory is also an omega limit set of a linear time-invariant system. |
It is often of interest to know which systems will approach a periodic trajectory when given a periodic input. Results are available for certain classes of systems, such as contracting systems, showing that they always entrain to periodic inputs. In contrast to this, we demonstrate that there exist systems which are globally exponentially stable yet do not entrain to a periodic input. This could be seen as surprising, as it is known that globally exponentially stable systems are in fact contracting with respect to some Riemannian metric. The paper also addresses the broader issue of entrainment when an input is added to a contractive system. |
The identification of constraints on system parameters that will ensure that a system achieves desired requirements remains a challenge in synthetic biology, where components unintendedly affect one another by perturbing the cellular environment in which they operate. This paper shows how to solve this problem optimally for a class of input/output system-level specifications, and for unintended interactions due to resource sharing. Specifically, we show how to solve the problem based on the input/output properties of the subsystems and on the unintended interaction map. Our approach is based on the elimination of quantifiers in monotone properties of the system. We illustrate applications of this methodology to guaranteeing system-level performance of multiplexed and sequential biosensing and of bistable genetic circuits. |
Steady state non-monotonic ("biphasic") dose responses are often observed in experimental biology, which raises the control theoretic question of identifying which possible mechanisms might underlie such behaviors. It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a non-monotonic response, and it has been informally conjectured that this condition is also necessary. However, this conjecture has been disproved with an example of a system in which input and output nodes are the same. In this paper, we show that the converse implication does hold when the input and output are distinct. Towards this aim, we give necessary and sufficient conditions for when minors of a symbolic matrix have mixed signs. Finally, we study in full generality when a model of immune T-cell activation could exhibit a steady state non-monotonic dose response. |
Motivated by the current interest in using Artificial intelligence (AI) tools in control design, this paper takes the first steps towards bridging results from gradient methods for solving the LQR control problem, and neural networks. More specifically, it looks into the case where one wants to find a Linear Feed-Forward Neural Network (LFFNN) that minimizes the Linear Quadratic Regulator (LQR) cost. This work develops gradient formulas that can be used to implement the training of LFFNNs to solve the LQR problem, and derives an important conservation law of the system. This conservation law is then leveraged to prove global convergence of solutions and invariance of the set of stabilizing networks under the training dynamics. These theoretical results are then followed by and extensive analysis of the simplest version of the problem (the ``scalar case'') and by numerical evidence of faster convergence of the training of general LFFNNs when compared to traditional direct gradient methods. These results not only serve as indication of the theoretical value of studying such a problem, but also of the practical value of LFFNNs as design tools for data-driven control applications. |
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