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Feedback cycles

The amisc library handles multidisciplinary systems with feedback coupling using a fixed-point iteration (FPI) routine. This approach is crucial for solving systems where the outputs of some components are inputs to others, and vice versa, forming cycles of dependencies.

Feedback coupling

In multidisciplinary systems, feedback coupling occurs when the outputs of one component influence the inputs of another and vice versa, forming cycles. These cycles can be represented as strongly-connected components (SCCs) in a graph-theoretic representation of the system. Each SCC represents a group of components that are interdependent.

Example

In the following system of 3 nonlinear equations: \(f_1, f_2, f_3\), the \((f_2, f_3)\) group forms an SCC with coupling variables \((y_2, y_3)\). Viewing the condensation of the system graph using the networkx library highlights this feedback cycle. 

from amisc import System
import networkx as nx

def f1(x):
    y1 = x ** 2
    return y1

def f2(y1, y3):
    y2 = y1 + np.sin(y3)
    return y2

def f3(y2):
    y3 = np.cos(y2)
    return y3

system = System(f1, f2, f3)
dag = nx.condensation(system.graph())  # Group SCCs to form a directed-acyclic-graph

for idx, node in dag.nodes.items():
    print(node['members'])

# gives 'f1' and ['f2', 'f3']

Fixed-Point Iteration (FPI)

An FPI routine is employed to solve for the coupling variables within each SCC. The process involves iteratively updating the coupling variables until convergence is achieved. The steps are as follows:

  1. Initialization - Begin with an initial guess for the coupling variables \(\xi^{(0)}\) within the SCC.

  2. Iterative update - For each iteration \( i \), update the coupling variables \(\xi^{(i)}\) using the function \( F(\xi^{(i-1)}) \), which evaluates the component models in the SCC using the current estimates of the coupling variables: \begin{equation} \xi^{(i)} = F(\xi^{(i-1)}). \end{equation}

  3. Convergence check - Continue iterating until the change in the coupling variables is below a specified tolerance, i.e., \( \lVert\xi^{(i)} - \xi^{(i-1)}\rVert < \eta \), where \(\eta\) is the convergence tolerance.

Usage in amisc

When using amisc surrogates for prediction, the system's graph structure is divided into SCCs, and each SCC is solved independently using the FPI routine with the surrogate models. The feedback dependencies are automatically managed under-the-hood during prediction, and amisc allows the user to specify some keywords to fine-tune the FPI routine, including the convergence tolerance \(\eta\) and the maximum number of iterations.

Example

Using the three-component system of \((f_1, f_2, f_3)\) defined above, predictions are obtained with the surrogate predict() method, with FPI options passed as keywords. 

system = System(f1, f2, f3)  # defined in previous example

system.predict({'x': 0.5}, max_fpi_iter=100, fpi_tol=1e-10, andersom_mem=5)

Note that the System implements the Anderson acceleration algorithm for enhancing the convergence of FPI.