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constrained nonlinear optimization for scientific machine learning, UQ, and AI

Project description

About Mystic

The mystic framework provides a collection of optimization algorithms and tools that allows the user to more robustly (and easily) solve hard optimization problems. All optimization algorithms included in mystic provide workflow at the fitting layer, not just access to the algorithms as function calls. mystic gives the user fine-grained power to both monitor and steer optimizations as the fit processes are running. Optimizers can advance one iteration with Step, or run to completion with Solve. Users can customize optimizer stop conditions, where both compound and user-provided conditions may be used. Optimizers can save state, can be reconfigured dynamically, and can be restarted from a saved solver or from a results file. All solvers can also leverage parallel computing, either within each iteration or as an ensemble of solvers.

Where possible, mystic optimizers share a common interface, and thus can be easily swapped without the user having to write any new code. mystic solvers all conform to a solver API, thus also have common method calls to configure and launch an optimization job. For more details, see mystic.abstract_solver. The API also makes it easy to bind a favorite 3rd party solver into the mystic framework.

Optimization algorithms in mystic can accept parameter constraints, as “soft constraints” (i.e. penalties, which “penalize” regions of solution space that violate the constraints), or as “hard constraints” (i.e. constraints, which constrain the solver to only search in regions of space where the constraints are respected), or both. mystic provides a large selection of constraints, including probabistic and dimensionally reducing constraints. By providing a robust interface designed to enable the user to easily configure and control solvers, mystic greatly reduces the barrier to solving hard optimization problems.

Sampling, interpolation, and statistics in mystic are all designed to seamlessly couple with constrained optimization to facilitate scientific machine learning, uncertainty quantification, adaptive sampling, nonlinear interpolation, and artificial intelligence. mystic can convert systems of equalities and inequalities to hard or soft constraints using methods in mystic.symbolic. With mystic.constraints.vectorize, constraints can be converted to kernel transforms for use in machine learning. Similarly, mystic provides tools for accurately producing emulators on an irregular grid using mystic.math.interpolate, which includes methods for solving for gradients and Hessians. mystic.samplers use optimizers to drive adaptive sampling toward the first and second order critical points of the response surface, yielding highly-informative training data sets and ensuring emulator accuracy. mystic.math.discrete defines constrained discrete probability measures, which can be used in constrained statistical optimization and learning.

mystic is in active development, so any user feedback, bug reports, comments, or suggestions are highly appreciated. A list of issues is located at, with a legacy list maintained at

Major Features

mystic provides a stock set of configurable, controllable solvers with:

  • a common interface

  • a control handler with: pause, continue, exit, and callback

  • ease in selecting initial population conditions: guess, random, etc

  • ease in checkpointing and restarting from a log or saved state

  • the ability to leverage parallel & distributed computing

  • the ability to apply a selection of logging and/or verbose monitors

  • the ability to configure solver-independent termination conditions

  • the ability to impose custom and user-defined penalties and constraints

To get up and running quickly, mystic also provides infrastructure to:

  • easily generate a model (several standard test models are included)

  • configure and auto-generate a cost function from a model

  • configure an ensemble of solvers to perform a specific task

Current Release

The latest released version of mystic is available from:

mystic is distributed under a 3-clause BSD license.

Development Version

You can get the latest development version with all the shiny new features at:

If you have a new contribution, please submit a pull request.


mystic can be installed with pip:

$ pip install mystic

To include optional scientific Python support, with scipy, install:

$ pip install mystic[math]

To include optional plotting support with matplotlib, install:

$ pip install mystic[plotting]

To include optional parallel computing support, with pathos, install:

$ pip install mystic[parallel]


mystic requires:

  • python (or pypy), >=3.8

  • setuptools, >=42

  • cython, >=0.29.30

  • numpy, >=1.0

  • sympy, >=0.6.7

  • mpmath, >=0.19

  • dill, >=0.3.8

  • klepto, >=0.2.5

Optional requirements:

  • matplotlib, >=0.91

  • scipy, >=0.6.0

  • pathos, >=0.3.2

  • pyina, >=0.2.8

More Information

Probably the best way to get started is to look at the documentation at Also see mystic.tests for a set of scripts that demonstrate several of the many features of the mystic framework. You can run the test suite with python -m mystic.tests. There are several plotting scripts that are installed with mystic, primary of which are mystic_log_reader (also available with python -m mystic) and the mystic_model_plotter (also available with python -m mystic.models). There are several other plotting scripts that come with mystic, and they are detailed elsewhere in the documentation. See for examples that demonstrate the basic use cases for configuration and launching of optimization jobs using one of the sample models provided in mystic.models. Many of the included examples are standard optimization test problems. The use of constraints and penalties are detailed in while more advanced features leveraging ensemble solvers, machine learning, uncertainty quantification, and dimensional collapse are found in The scripts in demonstrate leveraging pathos for parallel computing, as well as demonstrate some auto-partitioning schemes. mystic has the ability to work in product measure space, and the scripts in show how to work with product measures at a low level. The source code is generally well documented, so further questions may be resolved by inspecting the code itself. Please feel free to submit a ticket on github, or ask a question on stackoverflow (@Mike McKerns). If you would like to share how you use mystic in your work, please send an email (to mmckerns at uqfoundation dot org).

Instructions on building a new model are in mystic.models.abstract_model. mystic provides base classes for two types of models:

  • AbstractFunction [evaluates f(x) for given evaluation points x]

  • AbstractModel [generates f(x,p) for given coefficients p]

mystic also provides some convienence functions to help you build a model instance and a cost function instance on-the-fly. For more information, see mystic.forward_model. It is, however, not necessary to use base classes or the model builder in building your own model or cost function, as any standard Python function can be used as long as it meets the basic AbstractFunction interface of cost = f(x).

All mystic solvers are highly configurable, and provide a robust set of methods to help customize the solver for your particular optimization problem. For each solver, a minimal (scipy.optimize) interface is also provided for users who prefer to configure and launch their solvers as a single function call. For more information, see mystic.abstract_solver for the solver API, and each of the individual solvers for their minimal functional interface.

mystic enables solvers to use parallel computing whenever the user provides a replacement for the (serial) Python map function. mystic includes a sample map in mystic.python_map that mirrors the behavior of the built-in Python map, and a pool in mystic.pools that provides map functions using the pathos (i.e. multiprocessing) interface. mystic solvers are designed to utilize distributed and parallel tools provided by the pathos package. For more information, see mystic.abstract_map_solver, mystic.abstract_ensemble_solver, and the pathos documentation at

Important classes and functions are found here:

  • mystic.solvers [solver optimization algorithms]

  • mystic.termination [solver termination conditions]

  • mystic.strategy [solver population mutation strategies]

  • mystic.monitors [optimization monitors]

  • mystic.symbolic [symbolic math in constraints]

  • mystic.constraints [constraints functions]

  • mystic.penalty [penalty functions]

  • mystic.collapse [checks for dimensional collapse]

  • mystic.coupler [decorators for function coupling]

  • mystic.pools [parallel worker pool interface]

  • mystic.munge [file readers and writers]

  • mystic.scripts [model and convergence plotting]

  • mystic.samplers [optimizer-guided sampling]

  • [hypercube measure support plotting]

  • mystic.forward_model [cost function generator]

  • [constraints, wrappers, and other tools]

  • mystic.cache [results caching and archiving]

  • mystic.models [models and test functions]

  • mystic.math [mathematical functions and tools]

Important functions within mystic.math are found here:

  • mystic.math.Distribution [a sampling distribution object]

  • mystic.math.legacydata [classes for legacy data observations]

  • mystic.math.discrete [classes for discrete measures]

  • mystic.math.measures [tools to support discrete measures]

  • mystic.math.approx [tools for measuring equality]

  • mystic.math.grid [tools for generating points on a grid]

  • mystic.math.distance [tools for measuring distance and norms]

  • mystic.math.poly [tools for polynomial functions]

  • mystic.math.samples [tools related to sampling]

  • mystic.math.integrate [tools related to integration]

  • mystic.math.interpolate [tools related to interpolation]

  • mystic.math.stats [tools related to distributions]

Solver, Sampler, and model API definitions are found here:

  • mystic.abstract_sampler [the sampler API definition]

  • mystic.abstract_solver [the solver API definition]

  • mystic.abstract_map_solver [the parallel solver API]

  • mystic.abstract_ensemble_solver [the ensemble solver API]

  • mystic.models.abstract_model [the model API definition]

mystic also provides several convience scripts that are used to visualize models, convergence, and support on the hypercube. These scripts are installed to a directory on the user’s $PATH, and thus can be run from anywhere:

  • mystic_log_reader [parameter and cost convergence]

  • mystic_log_converter [logfile format converter]

  • mystic_collapse_plotter [convergence and dimensional collapse]

  • mystic_model_plotter [model surfaces and solver trajectory]

  • support_convergence [convergence plots for measures]

  • support_hypercube [parameter support on the hypercube]

  • support_hypercube_measures [measure support on the hypercube]

  • support_hypercube_scenario [scenario support on the hypercube]

Typing --help as an argument to any of the above scripts will print out an instructive help message.


If you use mystic to do research that leads to publication, we ask that you acknowledge use of mystic by citing the following in your publication:

M.M. McKerns, L. Strand, T. Sullivan, A. Fang, M.A.G. Aivazis,
"Building a framework for predictive science", Proceedings of
the 10th Python in Science Conference, 2011;

Michael McKerns, Patrick Hung, and Michael Aivazis,
"mystic: highly-constrained non-convex optimization and UQ", 2009- ;

Please see or for further information.

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