| Publications by Eduardo D. Sontag in year 2026 |
| Books and proceedings |
| Articles in journal or book chapters |
| Metastatic melanoma presents a formidable challenge in oncology due to its high invasiveness and resistance to current treatments. Central to its ability to metastasize is the Notch signaling pathway, which, when activated through direct cell-cell interactions, propels cells into a metastatic state through mechanisms akin to the epithelial-mesenchymal transition (EMT). While the upregulation of miR-222 has been identified as a critical step in this metastatic progression, the mechanism through which this upregulation persists in the absence of active Notch signaling remains unclear. Here we introduce a dynamical system model that integrates miR-222 gene regulation with histone feedback mechanisms. Through computational analysis, we delineate the non-linear decision boundaries that govern melanoma cell fate transitions, taking into account the dynamics of Notch signaling and the role of epigenetic modifications. Our approach highlights the critical interplay between Notch signaling pathways and epigenetic regulation in dictating the fate of melanoma cells. |
| This work explores the integration of machine learning (ML) and mechanistic models (MM). While ML has demonstrated remarkable success in data-driven modeling across engineering, biology, and other scientific fields, MM remain essential for their interpretability and capacity to extrapolate beyond observed conditions based on established principles such as chemical kinetics and physiological processes. However, MM can be labor-intensive to construct and often rely on simplifying assumptions that may not fully capture real-world complexity. It is thus desirable to combine MM and ML approaches so as to enable more robust predictions, enhanced system insights, and improved handling of sparse or noisy data. A key challenge when doing so is ensuring that ML components do not disregard mechanistic information, potentially leading to overfitting or reduced interpretability. To address that challenge, this paper introduces the idea of Partially Uncertain Model Structures (PUMS) and investigates conditions that discourage the ML components from ignoring mechanistic constraints. This work also introduces the concept of embedded Physics-Informed Neural Networks (ePINNs), which consist of two loss-sharing neural networks that seamlessly blend ML and MM components. This work arose in the study of the context problem in synthetic biology. Engineered genetic circuits may exhibit unexpected behavior in living cells due to resource sharing. To illustrate the advantages of the ePINNs approach, this paper applies the framework to a gene network model subject to resource competition, demonstrating the effectiveness of this hybrid modeling approach in capturing complex system interactions while maintaining physical consistency. |
| Cancer therapies often fail when intolerable toxicity or drug-resistant cancer cells undermine otherwise effective treatment strategies. Over the past decade, adaptive therapy has emerged as a promising approach to postpone emergence of resistance by altering dose timing based on tumor burden thresholds. Despite encouraging results, these protocols often overlook the crucial role of toxicity-induced treatment breaks, which may permit tumor regrowth. Herein, we explore the following question: would toxicity feedback improve or hinder the efficacy of adaptive therapy? To address this question, we propose a mathematical framework for incorporating toxic feedback into treatment design. We find that the degree of competition between sensitive and resistant populations, along with the growth rate of resistant cells, critically modulates the impact of toxicity feedback on time to progression. Further, our model identifies circumstances where strategic treatment breaks, which may be based on either tumor size or toxicity, can mitigate overtreatment and extend time to progression, both at the baseline parameterization and across a heterogeneous virtual population. Taken together, these findings highlight the importance of integrating toxicity considerations into the design of adaptive therapy. |
| This paper investigates how the compositional structure of neural networks shapes their optimization landscape and training dynamics. We analyze the gradient flow associated with overparameterized optimization problems, which can be interpreted as training a neural network with linear activations. Remarkably, we show that the global convergence properties can be derived for any cost function that is proper and real analytic. We then specialize the analysis to scalar-valued cost functions, where the geometry of the landscape can be fully characterized. In this setting, we demonstrate that key structural features -- such as the location and stability of saddle points -- are universal across all admissible costs, depending solely on the overparameterized representation rather than on problem-specific details. Moreover, we show that convergence can be arbitrarily accelerated depending on the initialization, as measured by an imbalance metric introduced in this work. Finally, we discuss how these insights may generalize to neural networks with sigmoidal activations, showing through a simple example which geometric and dynamical properties persist beyond the linear case. |
| Conference articles |
| This paper studies gradient dynamics subject to additive stochastic noise, which may arise from sources such as stochastic gradient estimation, measurement noise, or stochastic sampling errors. To analyze the robustness of such stochastic gradient systems, the concept of small-covariance noise-to-state stability (NSS) is introduced, along with a Lyapunov-based characterization. Furthermore, the classical Polyak–Lojasiewicz (PL) condition on the objective function is generalized to the $\mathcal{K}$-PL condition via comparison functions, thereby extending its applicability to a broader class of optimization problems. It is shown that the stochastic gradient dynamics exhibit small-covariance NSS if the objective function satisfies the $\mathcal{K}$-PL condition and possesses a globally Lipschitz continuous gradient. This result implies that the trajectories of stochastic gradient dynamics converge to a neighborhood of the optimum with high probability, with the size of the neighborhood determined by the noise covariance. Moreover, if the $\mathcal{K}$-PL condition is strengthened to a $\mathcal{K}_\infty$-PL condition, the dynamics are NSS; whereas if it is weakened to a general positive-definite-PL condition, the dynamics exhibit integral NSS. The results further extend to objectives without globally Lipschitz gradients through appropriate step-size tuning. The proposed framework is further applied to the robustness analysis of policy optimization for the linear quadratic regulator (LQR) and logistic regression. |
| In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations. |
| In several applications, including synthetic biology, one often has input/output data on a system composed of many modules, and although the modules’ input/output functions and signals may be unknown, knowledge of the composition architecture can significantly reduce the amount of training data required to learn the system’s input/output mapping. Learning the modules’ input/output functions is also necessary for designing new systems from different composition architectures. Here, we propose a modular learning framework that incorporates prior knowledge of the system’s compositional structure to (a) identify the composing modules’ input/output functions from the system’s input/output data and (b) achieve this using a reduced amount of data compared to what would be required without knowledge of the compositional structure. To achieve this, we introduce the notion of modular identifiability, which allows recovery of modules’ input/output functions from a subset of the system’s input/output data, and we provide theoretical guarantees on a class of systems motivated by genetic circuits. We demonstrate the theory in computational studies showing that a neural network (NNET) that accounts for the compositional structure can learn the composing modules’ input/output functions and predict the system’s output on inputs outside of the training set distribution. By reducing the need for experimental data and enabling modules’ identification, this framework offers the potential to ease the design of synthetic biological circuits and of multi-module systems more generally. |
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