7,583 research outputs found
Data-driven PDE discovery with evolutionary approach
The data-driven models allow one to define the model structure in cases when
a priori information is not sufficient to build other types of models. The
possible way to obtain physical interpretation is the data-driven differential
equation discovery techniques. The existing methods of PDE (partial derivative
equations) discovery are bound with the sparse regression. However, sparse
regression is restricting the resulting model form, since the terms for PDE are
defined before regression. The evolutionary approach described in the article
has a symbolic regression as the background instead and thus has fewer
restrictions on the PDE form. The evolutionary method of PDE discovery (EPDE)
is described and tested on several canonical PDEs. The question of robustness
is examined on a noised data example
Unnatural Selection: A new formal approach to punctuated equilibrium in economic systems
Generalized Darwinian evolutionary theory has emerged as central to the description of economic process (e.g., Aldrich et. al., 2008). Here we demonstrate that, just as Darwinian principles provide necessary, but not sufficient, conditions for understanding the dynamics of social entities, in a similar manner the asymptotic limit theorems of information theory provide another set of necessary conditions that constrain the evolution of socioeconomic process. These latter constraints can, however, easily be formulated as a statistics-like analytic toolbox for the study of empirical data that is consistent with a generalized Darwinism, and this is no small thing
Differentiable Genetic Programming
We introduce the use of high order automatic differentiation, implemented via
the algebra of truncated Taylor polynomials, in genetic programming. Using the
Cartesian Genetic Programming encoding we obtain a high-order Taylor
representation of the program output that is then used to back-propagate errors
during learning. The resulting machine learning framework is called
differentiable Cartesian Genetic Programming (dCGP). In the context of symbolic
regression, dCGP offers a new approach to the long unsolved problem of constant
representation in GP expressions. On several problems of increasing complexity
we find that dCGP is able to find the exact form of the symbolic expression as
well as the constants values. We also demonstrate the use of dCGP to solve a
large class of differential equations and to find prime integrals of dynamical
systems, presenting, in both cases, results that confirm the efficacy of our
approach
Data-driven discovery of coordinates and governing equations
The discovery of governing equations from scientific data has the potential
to transform data-rich fields that lack well-characterized quantitative
descriptions. Advances in sparse regression are currently enabling the
tractable identification of both the structure and parameters of a nonlinear
dynamical system from data. The resulting models have the fewest terms
necessary to describe the dynamics, balancing model complexity with descriptive
ability, and thus promoting interpretability and generalizability. This
provides an algorithmic approach to Occam's razor for model discovery. However,
this approach fundamentally relies on an effective coordinate system in which
the dynamics have a simple representation. In this work, we design a custom
autoencoder to discover a coordinate transformation into a reduced space where
the dynamics may be sparsely represented. Thus, we simultaneously learn the
governing equations and the associated coordinate system. We demonstrate this
approach on several example high-dimensional dynamical systems with
low-dimensional behavior. The resulting modeling framework combines the
strengths of deep neural networks for flexible representation and sparse
identification of nonlinear dynamics (SINDy) for parsimonious models. It is the
first method of its kind to place the discovery of coordinates and models on an
equal footing.Comment: 25 pages, 6 figures; added acknowledgment
- …