Systems and methods for using deep machine learning for evolving boundary condition problems

A method for determining and providing value information and/or boundary activation information for a system defined in an at least partially bounded domain and having boundary conditions imposed at a boundary of the bounded domain, the method comprising:

• determining, by one or more processors, a plurality of paths for the system through at least a portion of the bounded domain based on an initial set of underlying features and a path evolution determined for each path, each path corresponding to a set of times comprising an initial time, a final time, and one or more intermediate times between the initial time and the final time, wherein the set of underlying features corresponds to respective financial values of a set of assets;
• causing, by the one or more processors, a deep backward stochastic differential equation (deep BSDE) solver to be trained until a convergence requirement is satisfied by:
• for each time step and for each path:
• determining the set of underlying features and a value corresponding to the set of underlying features,
• determining, based on the set of underlying features, whether a boundary condition has been activated for the path, and
• when it is determined that the boundary condition has been activated for a first time for the path, updating an output value for the path based on the boundary condition and an output time based on the time step at which the boundary condition was activated for the path,
• when the boundary condition was not activated for the path prior to the path reaching the final time, updating the output value for the path based on a final value corresponding to the set of underlying features at the final time,
• defining a set of output values comprising the output value for each path and determining one or more statistical measures of spread based on the set of output values, and
• modifying one or more parameters of a DNN of the deep BSDE solver based on the one or more statistical measures of spread;
• after the convergence requirement is satisfied, determining, by the one or more processors, at least one of (a) value information comprising a value corresponding to the system at the initial time and/or at one or more intermediate times based on the values and features for each path or (b) boundary activation information comprising a likelihood of boundary activation;
• causing, by the one or more processors, at least a portion of the at least one of (a) value information or (b) boundary activation information to be provided such that a user computing device receives the at least a portion of the at least one of (a) value information or (b) boundary activation information and provides a representation of the at least a portion of the at least one of (a) value information or (b) boundary activation information via an interactive user interface provided via a display of the user computing device;
• and
• causing, by the one or more processors, purchase of a portfolio comprising the set of assets based on the at least a portion of the at least one of (a) value information or (b) boundary activation information.