Bachir El Khadir

Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert

  • Ryan Cory-Wright
  • Cristina Cornelio
  • Sanjeeb Dash
  • Bachir El Khadir
  • Lior Horesh
    • Categories Basic Sciences Applications, Global Optimization Applications, Semi-definite Programming Tags , , ,

    The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in … Read more

    Learning Dynamical Systems with Side Information

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Convex Optimization, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or … Read more

    On Sum of Squares Representation of Convex Forms and Generalized Cauchy-Schwarz Inequalities

  • Bachir El Khadir
    • Categories Convex Optimization, Global Optimization Theory, Semi-definite Programming Tags , ,

    A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of … Read more

    Time-Varying Semidefinite Programs

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Basic Sciences Applications, Infinite Dimensional Optimization, Semi-definite Programming Tags , , , , ,

    We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data varies polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value … Read more

    A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , , ,

    We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally. CitationSubmitted for publicationArticleDownload View PDF

    On Algebraic Proofs of Stability for Homogeneous Vector Fields

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares … Read more