ANSR is an asynchronous neuro-evolutionary method for symbolic regression in which mathematical expressions are represented by feedforward neural networks (see Figure below). It combines evolutionary search over network master topology with continuous gradient-based tuning of model parameters.
Unlike standard neuro-evolutionary approaches, ANSR does not judge new offspring only by their immediate performance after mutation or crossover. Since structural changes often require additional coefficient tuning, promising models may otherwise be discarded too early. ANSR addresses this with lifelong asynchronous refinement and maturity-based replacement: offspring are allowed to develop in parallel with the population and are compared with individuals at a similar stage of refinement. This makes the search more patient, preserves structurally novel candidates longer, and better separates topology quality from short-term parameter readiness.
The ros2/ folder wraps ANSR as a ROS 2 Jazzy module for the
CoreSense ecosystem: learning runs as a cancellable
action, best-so-far model retrieval as a service, per-session workspaces, and
mid-run training-data swaps. The core in source_code/ is used as-is, run as
a supervised subprocess. See ros2/README.md for the
interfaces, build steps, and tests.
Candidate expressions are represented as neural networks composed of layers of units, where each unit represents a mathematical operation (e.g., addition, multiplication, sine, cosine). The master topology defines the structure of the individuals. It specifies the number of layers and the types of units used in each layer. Additionally, identity connections between layers can be enabled, allowing layers to be skipped.
The method maintains three populations of individuals: mainpop, offsprings, and archive. The first two populations have fixed size, whereas archive is not size-limited. The mainpop contains individuals that are used to generate new individuals via crossover and mutation. They are initially generated randomly based on the specified master topology. The offsprings population contains newly created individuals. The archive contains non-dominated solutions among all solutions generated up to the given point in the computation. Individuals in this population do not actively participate in the optimization process.
In ANSR, individuals are regularly refined via backpropagation in batches, where the number of optimization iterations is a method parameter. They are kept in the population until it becomes evident that an offspring individual should be promoted to the main population. An individual from either the main population or the offspring population is discarded when it no longer shows potential for further improvement, according to the replacement rules.
The individuals are represented as neural networks composed of layers of units. Each unit represents a mathematical operation.
The following units are implemented in the SRUnits.py file:
UnitMultiply: multiplicationUnitDivide: divisionUnitReciprocal: reciprocalUnitSin: sineUnitCos: cosineUnitTanh: hyperbolic tangentUnitArcTan: arctangentUnitSquare: squareUnitSqrt: square rootUnitCube: cubeUnitSign: signUnitIdent: identity
One line in the file represents one layer, followed by another line specifying whether skip connections are allowed. In the output layer, there must always be exactly one identity unit.
Example master topology file:
UnitMultiply:4, UnitIdent:4 # 1st hidden layer with 4 multiplication and 4 identity units
identities:True # allow skip connections from the input to this layer
UnitMultiply:4, UnitIdent:4 # 2nd hidden layer with 4 multiplication and 4 identity units
identities:true # allow skip connections from the input and the 1st hidden layer to this layer
UnitIdent:1 # output layer with a single identity unit
identities:false # do not allow skip connections
The example master topology defines a network with 2 hidden layers and 1 output neuron which is a simple identity. The hidden layers consist of 4 multiplication units and 4 identity units. Skip connections are allowed from the input to both hidden layers, and from the first hidden layer to the second hidden layer.
Constraints can be defined in a JSON file. Once specified, a predefined number of data points is generated, and the RMSE of the constraints is computed on these points.
Available constraints types
constraint_gtvalue: The function output must be greater than a specified value.constraint_ltvalue: The function output must be less than a specified value.constraint_exactvalue: The function output must be equal to a specified value.constraint_concave_down: The function must be concave down with respect to a specified variable.constraint_decreasing: The function must be monotonically decreasing with respect to a specified variable.constraint_decreasing_positive_second_derivative: The function must be decreasing and must also have a positive second derivative with respect to a specified variable.constraint_increasing: The function must be monotonically increasing with respect to a specified variable.constraint_invariant: The function output must be equal to a specified state variable.constraint_resistors_diagonal: The function output on the diagonal must be half of the partial resistor.constraint_resistors_ltinput: The function output must be less than either of the two inputs.constraint_symmetry2vars: The function must be symmetric with respect to two specified variables.constraint_odd_symmetry: The function must be odd-symmetric with respect to a specified variable.
JSON file structure
The JSON file contains a list of constraints, each defined by its parameters:
class: The type of constraint (from the list above).weight: The relative importance of the constraint in the overall evaluation.domain: The domain used to generate sample points for evaluating the constraint.nbOfSamples: The number of sample points to generate for this constraint.args: Additional arguments required by the constraint.
Snippet of the JSON file, that defines two constraints - constraint_gtvalue and constraint_resistors_diagonal:
{
"constraints":
[
{
"class": "constraint_gtvalue",
"weight": 1.0,
"domain": [[[0.001, 40], [0.001, 40]]],
"nbOfSamples": 50,
"args": {
"test_value": 0.0
}
},
{
"class": "constraint_resistors_diagonal",
"weight": 1.0,
"domain": [[[0.001, 40], [0.001, 40]]],
"nbOfSamples": 50,
"args": {}
}
]
}
Navigate to the source_code directory and run the following commands.
Windows
python -m venv .venv creates a virtual environment
.venv\Scripts\activate activates the virtual environment
python -m pip install --upgrade pip
pip install -r requirements.txt installs all required packages
Linux
python3 -m venv .venv
source .venv/bin/activate
python3 -m pip install --upgrade pip
pip install -r requirements.txt
From the source_code directory, run the program with the following arguments:
- (*)
-t: master topology file -
-c: file with constraints - (*)
--train_data: data to use for training - (*)
--valid_data: data to use for validation - (*)
--main_pop_size: size of the main population - (*)
--offspring_pop_size: size of the offspring population -
--backprop_epoch: number of backpropagation epochs per iteration - (*)
-o: directory where to save results -
-s: seed
--addNewIndToOffspring: rules for adding new individuals to the offspring with two possible settingsaddNewIndToOffspring: add new individual to the offspring only if it is not fulladdNewIndToOffspringIfDominates: add new individual to the offspring if it is not full or if it dominates at least one individual in the offspring
--addOffspringToPopulation: rules for adding individual from the offspring to the mainpop with two possible settingsaddOffspringToPopulation: add individual from the offspring to the mainpop according to the rules described aboveaddOffspringToPopulationNever: never add individual from the offspring to the mainpop
--keepIndInMainPopulation: rules for keeping individual in the mainpop with two possible settingskeepIndInMainPopulationAlways: keep individual in the mainpop until it is replaced by individual from the offspringkeepIndInMainPopulationConditional: keep individual in the mainpop until it is replaced by individual from the offspring or dominated by other individual in the mainpop
--keepIndInOffspring: rules for keeping individual in the offspring with two possible settingskeepIndInOffspringAlways: keep individual in the offspring until it is moved to the mainpopkeepIndInOffspringConditional: keep individual in the offspring until it is moved to the mainpop or dominated by other individual in the offspring or its performance stopped improving
--networkActivityMutation: rules for network activity mutation with two possible settingsnetworkActivityMutation: mutate all elements with certain probabilitynetworkActivityMutationOnlyToInactive: mutate only active elements with certain probability
python Main.py -t topology_sin_cos_square_mult_2layers_2units.txt --train_data Nguyen-9c_n=100_train.csv --valid_data Nguyen-9c_n=100_valid.csv --main_pop_size 20 --offspring_pop_size 20 --maxTotalBackprops 100000 -o results/nguyen-9c -s 1
The output directory is organized hierarchically by seed, with each seed containing run metadata and saved individuals from both the archive and the main population.
- The base directory contains one subdirectory per seed:
seed=0seed=1- ...
Each seed directory contains:
-
archiveIndividuals/
Individuals belonging to the archive. -
mainPopulationIndividuals/
Individuals belonging to the mainpop. -
Run metadata and logs, such as:
configuration.txt: the configuration parameters used for the runinitial_population_phase_I.txt: the initial mainpop after first phase of tuninginitial_population_phase_II.txt: the initial mainpop after second phase of tuninglog.txt: log of the runpopulation.txt: the mainpop, archive, and offspring populations during several snapshots of the run
Both archiveIndividuals/ and mainPopulationIndividuals/ contain additional subdirectories that represent series of snapshots generated by one of the following quantities:
-
Number of backpropagations used
Example for backprop interval 1000 iterations:backprops_5400/,backprops_6400/,backprops_7400/, ... -
Number of generated individuals
Example for individual interval 100 ID:ids_100/,ids_200/,ids_300/, ... -
Elapsed time
Example for time interval 60s:time_60/,time_180/,time_240/, ...
These snapshot directories store the corresponding individuals at that stage of the run.
Each snapshot directory contains saved individuals, exported in:
- MATLAB format:
.m - Plain text format:
.txt
results/
└── basic-polynom/
├── seed=0/
│ ├── archiveIndividuals/
│ │ ├── backprops_6400/
│ │ ├── backprops_7400/
│ │ ├── ids_100/
│ │ ├── ids_120/
│ │ ├── time_180/
│ │ └── time_301/
│ ├── mainPopulationIndividuals/
│ │ ├── backprops_6400/
│ │ ├── ids_100/
│ │ └── time_180/
│ ├── configuration.txt
│ ├── error.err
│ ├── initial_population_phase_I.txt
│ ├── initial_population_phase_II.txt
│ ├── log.txt
│ ├── output.out
│ └── population.txt
├── seed=1/
├── seed=2/
├── seed=3/
└── ...
If you find this useful, please cite the paper:
@inproceedings{kubalik2026ansr,
author = {Kubal{\'\i}k, Ji{\v{r}}{\'\i} and Fu{\v{c}}elov{\'a}, Na{\v{d}}a and Babu{\v{s}}ka, Robert},
title = {Asynchronous Neuro-Evolutionary Symbolic Regression with Maturity-Based Replacement},
booktitle = {Genetic and Evolutionary Computation Conference (GECCO '26)},
year = {2026},
doi = {10.1145/3795095.3805164}
}
This work has received funding from the European Union's Horizon Europe research and innovation programme under grant agreement No 101070254 (CORESENSE). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the granting authority can be held responsible for them.