Modulus Sym uses constraints to define the objectives for neural network training. These house a set of nodes from which a computational graph is built for execution as well as loss function. Many physical problems require multiple training objectives/constraints to be defined in a well-posed manner. The constraints in Modulus Sym are designed to provide the means for intuitively setting up multi-objective problems.

Several types of constraints are available within Modulus Sym that allow you to quickly setup your AI training either in a physics-informed or data-informed fashion.
At the core, the various constraints in Modulus Sym sample a dataset, execute the computational nodes on the generated samples and compute the loss for each constraint. This individual loss is
then combined with the losses of other user-defined constraints using a aggregator method selected. The combined loss is then passed to the optimizer for
optimization. The different variants available in Modulus Sym makes the definition of some common types of constraints easy so that you do not have to write a lot of boilerplate code
for sampling and evaluating. Each constraint is recorded in the `Domain`

class which is input to the `Solver`

.

The word “continuous” here is used primarily to indicate the constraints is applied on points sampled uniformly randomly in the continuous space or surface of the geometry. For a physics-informed training, it is very typical to apply the PDE constraints in the interior of the domain and boundary conditions on the domain boundaries. Several other constraints to apply integral losses are also available.

### PointwiseBoundaryConstraint

The boundary of a Modulus Sym’ geometry object can be sampled using `PointwiseBoundaryConstraint`

class.
This will sample the entire boundary of the geometry specified as input to the `geometry`

parameter.
In the case of 1D, the boundaries are the end points, for 2D, its the points along the perimeter,
and for 3D its the points on the surface of the geometry.

Mathematically the pointwise boundary constraint can be represented as

By default, all the boundaries will be sampled by this class and subsampling is possible using the `criteria`

parameter.
The `outvar`

parameter is used for describing the constraint. The outvar dictionaries are used when
unrolling the computational graph (specified using the `nodes`

parameter) and computing the loss.
The number of points to sample on each boundary are specified using the `batch_size`

parameter.
A detailed description of all the arguments can be found in the API documentation.

Mathematically the pointwise boundary constraint can be represented as

(134)\[L = \left| \int_{\partial \Omega} ( u_{net}(x,y,z) - \phi ) \right|^p = \left| \frac{S}{B} \sum_{i}(u_{net}(x_i, y_i, z_i) - \phi) \right|^p\]

Where \(L\) is the loss, \(\partial \Omega\) is the boundary, \(u_{net}(x,y,z)\) is the network prediction for the keys in `outvar`

,
\(\phi\) is the value specified in the `outvar`

and \(p\) is the norm of the loss. \(S\) and \(B\) are the surface area/perimeter and batch size respectively.

Below, a simple boundary condition definition is shown. Here the problem is trying to only satisfy the boundary.

Listing 11 *Boundary constraint*

```
```import numpy as np
from sympy import Symbol, Function, Number, pi, sin
import modulus.sym
from modulus.sym.hydra import to_absolute_path, ModulusConfig
from modulus.sym.solver import Solver
from modulus.sym.domain import Domain
from modulus.sym.geometry.primitives_1d import Point1D, Line1D
from modulus.sym.domain.constraint import (
PointwiseBoundaryConstraint,
)
from modulus.sym.key import Key
from modulus.sym.node import Node
from modulus.sym.models.fully_connected import FullyConnectedArch
@modulus.main(config_path="conf", config_name="config")
def run(cfg: ModulusConfig) -> None:
*# make list of nodes to unroll graph on*
u_net = FullyConnectedArch(
input_keys=[Key("x")], output_keys=[Key("u")], nr_layers=3, layer_size=32
)
nodes = [u_net.make_node(name="u_network")]
*# add constraints to solver*
*# make geometry*
x = Symbol("x")
geo = Line1D(0, 1)
*# make domain*
domain = Domain()
*# bcs*
bc = PointwiseBoundaryConstraint(
nodes=nodes,
geometry=geo,
outvar={"u": 0},
batch_size=2,
)
domain.add_constraint(bc, "bc")
*# make solver*
slv = Solver(cfg, domain)
*# start solver*
slv.solve()
if __name__ == "__main__":
run()

### PointwiseInteriorConstraint

The interior of a Modulus Sym’ geometry object can be sampled using `PointwiseInteriorConstraint`

class.
This will sample the entire interior of the geometry specified as input to the `geometry`

parameter.

Similar to boundary sampling, subsampling is possible using the `criteria`

parameter. The `outvar`

and `batch_size`

parameters
work in the same way as `PointwiseBoundaryConstraint`

.
A detailed description of all the arguments can be found in the API documentation.

Mathematically the pointwise interior constraint can be represented as

(135)\[L = \left| \int_{\Omega} ( u_{net}(x,y,z) - \phi ) \right|^p = \left| \frac{V}{B} \sum_{i}(u_{net}(x_i, y_i, z_i) - \phi) \right|^p\]

Where \(L\) is the loss, \(\Omega\) is the interior, \(u_{net}(x,y,z)\) is the network prediction for the keys in `outvar`

,
\(\phi\) is the value specified in the `outvar`

and \(p\) is the norm of the loss. \(V\) and \(B\) are the volume/area and batch size respectively.

Below, a simple interior constraint definition is shown.

Listing 12 *Interior constraint*

```
```import numpy as np
from sympy import Symbol, Function, Number, pi, sin
import modulus.sym
from modulus.sym.hydra import to_absolute_path, ModulusConfig
from modulus.sym.solver import Solver
from modulus.sym.domain import Domain
from modulus.sym.geometry.primitives_1d import Point1D, Line1D
from modulus.sym.domain.constraint import (
PointwiseBoundaryConstraint,
PointwiseInteriorConstraint,
)
from modulus.sym.domain.inferencer import PointwiseInferencer
from modulus.sym.key import Key
from modulus.sym.node import Node
from modulus.sym.models.fully_connected import FullyConnectedArch
from modulus.sym.eq.pde import PDE
class CustomPDE(PDE):
def __init__(self, f=1.0):
*# coordinates*
x = Symbol("x")
*# make input variables*
input_variables = {"x": x}
*# make u function*
u = Function("u")(*input_variables)
*# source term*
if type(f) is str:
f = Function(f)(*input_variables)
elif type(f) in [float, int]:
f = Number(f)
*# set equations*
self.equations = {}
self.equations["custom_pde"] = (
u.diff(x, 2) - f
) *# "custom_pde" key name will be used in constraints*
@modulus.main(config_path="conf", config_name="config")
def run(cfg: ModulusConfig) -> None:
*# make list of nodes to unroll graph on*
eq = CustomPDE(f=1.0)
u_net = FullyConnectedArch(
input_keys=[Key("x")], output_keys=[Key("u")], nr_layers=3, layer_size=32
)
nodes = eq.make_nodes() + [u_net.make_node(name="u_network")]
*# add constraints to solver*
*# make geometry*
x = Symbol("x")
geo = Line1D(0, 1)
*# make domain*
domain = Domain()
*# interior*
interior = PointwiseInteriorConstraint(
nodes=nodes,
geometry=geo,
outvar={"custom_pde": 0},
batch_size=100,
bounds={x: (0, 1)},
)
domain.add_constraint(interior, "interior")
*# make solver*
slv = Solver(cfg, domain)
*# start solver*
slv.solve()
if __name__ == "__main__":
run()

### IntegralBoundaryConstraint

This constraint samples points on the boundary of the geometry object similar to the `PointwiseBoundaryConstraint`

, but now instead of computing a pointwise loss, it computes monte-carlo integration of specified variable and then assigns the specified value to it to compute the loss. Mathematically this can be shown as below:

(136)\[L = \left| \int_{\partial \Omega} u_{net}(x,y,z) - \phi \right|^p = \left| \left(\frac{S}{B} \sum_{i}u_{net}(x_i, y_i, z_i)\right) - \phi \right|^p\]

Where \(L\) is the loss, \(\partial \Omega\) is the boundary, \(u_{net}(x,y,z)\) is the network prediction for the keys in `outvar`

,
\(\phi\) is the value specified in the `outvar`

and \(p\) is the norm of the loss. \(S\) and \(B\) are the volume/area and batch size respectively.

Please note that the `batch_size`

has a slightly different meaning here. The `batch_size`

parameter is used to define the number of instances of integrals to apply while
the `integral_batch_size`

is the actual points sampled on the boundary.

Below, a simple integral constraint definition is shown.

Listing 13 *Integral constraint*

```
```import numpy as np
from sympy import Symbol, Function, Number, pi, sin
import modulus.sym
from modulus.sym.hydra import to_absolute_path, ModulusConfig
from modulus.sym.solver import Solver
from modulus.sym.domain import Domain
from modulus.sym.geometry.primitives_1d import Point1D, Line1D
from modulus.sym.domain.constraint import (
IntegralBoundaryConstraint,
)
from modulus.sym.domain.inferencer import PointwiseInferencer
from modulus.sym.key import Key
from modulus.sym.node import Node
from modulus.sym.models.fully_connected import FullyConnectedArch
from modulus.sym.eq.pde import PDE
@modulus.main(config_path="conf", config_name="config")
def run(cfg: ModulusConfig) -> None:
*# make list of nodes to unroll graph on*
u_net = FullyConnectedArch(
input_keys=[Key("x")], output_keys=[Key("u")], nr_layers=3, layer_size=32
)
nodes = [u_net.make_node(name="u_network")]
*# add constraints to solver*
*# make geometry*
x = Symbol("x")
geo = Line1D(0, 1)
*# make domain*
domain = Domain()
*# integral*
integral = IntegralBoundaryConstraint(
nodes=nodes,
geometry=geo,
outvar={"u": 0},
batch_size=1,
integral_batch_size=100,
)
domain.add_constraint(integral, "integral")
*# make solver*
slv = Solver(cfg, domain)
*# start solver*
slv.solve()
if __name__ == "__main__":
run()

For discrete constrains, the constraint is applied on a structure of fixed points taken from a discretized representation of the space. The simplest example of this is a uniform grid.

### SupervisedGridConstraint

This constraint performs standard supervised training on grid data. This constraint also supports the use of multiple workers, which are particularly important when using lazy loading. This constraint is primarily used for grid based models like Fourier Neural Operators. Losses computed in these constraint are pointwise similar to the above boundary and interior constraints.

Below, a simple supervised grid constraint definition is shown.

Listing 14 *Supervised Grid Constraint from the Darcy flow example*

```
```import modulus.sym
from modulus.sym.hydra import to_absolute_path, instantiate_arch, ModulusConfig
from modulus.sym.key import Key
from modulus.sym.solver import Solver
from modulus.sym.domain import Domain
from modulus.sym.domain.constraint import SupervisedGridConstraint
from modulus.sym.dataset import HDF5GridDataset
from modulus.sym.utils.io.plotter import GridValidatorPlotter
from utilities import download_FNO_dataset
@modulus.main(config_path="conf", config_name="config_FNO")
def run(cfg: ModulusConfig) -> None:
*# load training/ test data*
input_keys = [Key("coeff", scale=(7.48360e00, 4.49996e00))]
output_keys = [Key("sol", scale=(5.74634e-03, 3.88433e-03))]
download_FNO_dataset("Darcy_241", outdir="datasets/")
train_path = to_absolute_path(
"datasets/Darcy_241/piececonst_r241_N1024_smooth1.hdf5"
)
test_path = to_absolute_path(
"datasets/Darcy_241/piececonst_r241_N1024_smooth2.hdf5"
)
*# make datasets*
train_dataset = HDF5GridDataset(
train_path, invar_keys=["coeff"], outvar_keys=["sol"], n_examples=1000
)
test_dataset = HDF5GridDataset(
test_path, invar_keys=["coeff"], outvar_keys=["sol"], n_examples=100
)
*# make list of nodes to unroll graph on*
model = instantiate_arch(
input_keys=input_keys,
output_keys=output_keys,
cfg=cfg.arch.fno,
)
nodes = model.make_nodes(name="FNO", jit=cfg.jit)
*# make domain*
domain = Domain()
*# add constraints to domain*
supervised = SupervisedGridConstraint(
nodes=nodes,
dataset=train_dataset,
batch_size=cfg.batch_size.grid,
num_workers=4, *# number of parallel data loaders*
)
domain.add_constraint(supervised, "supervised")
*# make solver*
slv = Solver(cfg, domain)
*# start solver*
slv.solve()
if __name__ == "__main__":
run()

User defined custom constraints can be implemented by inheriting from the `Constraint`

class defined in `modulus/domain/constraint/constraint.py`

.
There are 3 methods you will need to specify to use your constraint, `load_data`

, `loss`

and `save_batch`

.
The `load_data`

method is used to load a mini-batch of data from the internal dataloader. The `loss`

method computes loss used when training.
Lastly, the `save_batch`

method specifies how to save a batch of for debugging or post processing.
This structure is meant to be general and allows for many complex constraints to be formed such as those used in variational methods.
For references on implementations of these methods please refer to any of the above base constraints.