Free-form discovery of differential equations

A system for modeling a dynamic physical system or a process comprising:

• at least one memory storage device;
• and
• one or more hardware processors operatively connected to said at least one memory storage device, said one or more hardware processors configured to:
• receive a training data set providing training data representing a phenomenon of a physical system to be modeled, said training data including input training data representing input conditions of the physical system or phenomenon and output training data relating to measurement outputs observed at the physical system or phenomenon;
• access a plurality of grammars stored in a memory storage device, a respective grammar encoding a set of rules and syntax constraints defining admissible relationships between entities comprising differential operators, primitive operators, variables, and coefficients representing a respective differential equation expression, each respective grammar used to construct a mapping of the corresponding differential equation expression to pre-defined data points in a defined multi-dimensional model space, each respective differential equation expression corresponding to a respective valid differential equation in the defined multi-dimensional model space;
• iteratively process said plurality of grammars to identify a simplest differential equation expression corresponding to the respective valid differential equation that relates the input training data and measurement outputs observed such that a numerical discrepancy between an evaluation of the simplest differential equation for the input training data and the measurement outputs observed is bounded within a numerical error bound tolerance, wherein to iteratively process said plurality of grammars, said one or more hardware processors are further configured to:
• search said defined multi-dimensional model space to determine candidate grammars corresponding to differential equation expressions having mapped pre-defined data points in the defined multi-dimensional model space that link the input training data and output data points relating to the measurement outputs observed, wherein to search, said one or more hardware processors are further configured to:
• generate a mixed integer non-linear program (MINLP) defining an objective function representing the defined multi-dimensional model space as an expression tree comprising operators, said operators including said differential operators, and operands, together with a corresponding companion expression tree structure to encode derivatives of the differential equation expression;
• and
• solve the MINLP by optimizing the objective function subject to one or more constraints to determine a model in the defined multi-dimensional model space that fits the observed measurement data while minimizing a complexity of the differential equation expression, said solving comprising:
• for a chosen candidate grammar:
• obtain a numerical evaluation for input data points of said input training data and the output data points of said output training data;
• determine a numerical discrepancy between an evaluation of the simplest differential equation expression for the input training data and the measurements outputs observed, said MINLP having integer decision parameters creating a functional form of a grammar, and numerically evaluating non-linear decision parameters of the grammar at the data points;
• and
• compare the numerical evaluation for input data points and output data points for the candidate grammar against a similar numerical evaluation based on input data points of said input training data and values of the measurement outputs observed;
• and
• based on said comparing, identify the simplest differential equation expression corresponding to a chosen candidate grammar whose numerical evaluation of the input training data provide output data points closest to said values of the measurements outputs observed;
• output the simplest differential equation expression satisfying the training data within said numerical error bound tolerance, and
• use said output simplest differential equation expression for modeling a behavior of the dynamic physical system to be modeled.

A computer program product for modeling a dynamic physical system or a process, the computer program product comprising a non-transitory storage medium readable by a processing circuit and storing instructions run by the processing circuit for performing a method comprising:

• receiving, at the processing circuit, a training data set providing training data representing a phenomenon or a physical system to be modeled, said training data including input training data representing input conditions of the physical system or phenomenon and output training data relating to measurement outputs observed at the physical system or phenomenon;
• accessing, using the processing circuit, a plurality of grammars stored in a memory storage device, a respective grammar encoding a set of rules and syntax constraints defining admissible relationships between entities comprising differential operators, primitive operators, variables, and coefficients representing a respective differential equation expression, each respective grammar used to construct a mapping of the corresponding differential equation expression to pre-defined data points in a defined multi-dimensional model space, each respective differential equation expression corresponding to a respective valid differential equation in the defined multi-dimensional model space;
• iteratively processing, at the processing circuit, said plurality of grammars to identify a simplest differential equation expression corresponding to a valid differential equation that relates the input training data and measurement outputs observed such that a numerical discrepancy between an evaluation of the simplest differential equation expression for the input training data and the measurement outputs observed is bounded within a numerical error bound tolerance, said iterative processing of the plurality of grammars comprising:
• searching said defined multi-dimensional model space to determine candidate grammars corresponding to differential equation expressions having mapped pre-defined data points in the defined multi-dimensional model space that link the input training data and output data points relating to the measurement outputs observed, wherein to search, said program instructions further configuring the computer to:
• generate a mixed integer non-linear program (MINLP) defining an objective function representing the defined multi-dimensional model space as an expression tree comprising operators, said operators including said differential operators, and operands, together with a corresponding companion expression tree structure to encode derivatives of the differential equation expression;
• and
• solve the MINLP by optimizing the objective function subject to one or more constraints to determine a model in the defined multi-dimensional model space that fits the observed measurement data while minimizing a complexity of the differential equation expression, said solving comprising:
• for a chosen candidate grammar:
• obtaining a numerical evaluation for input data points of said input training data and the output data points of said output training data;
• determining a numerical discrepancy between an evaluation of the simplest differential equation expression for the input training data and the measurements outputs observed, said MINLP having integer decision parameters creating a functional form of the grammar, and numerically evaluating non-linear decision parameters of the grammar at the data points;
• and
• comparing the numerical evaluation for input data points and output data points for the candidate grammar against a similar numerical evaluation based on input data points of said input training data and values of the measurement outputs observed;
• and
• based on said comparing, identifying the simplest differential equation expression corresponding to a chosen candidate grammar whose numerical evaluation of the input training data provide output data points closest to said values of the measurements outputs observed;
• outputting the simplest differential equation expression satisfying the data within said numerical error bound tolerance, and
• using, by said hardware processor, said output simplest differential equation expression for modeling a behavior of the dynamic physical system to be modeled.