Table of Contents

Class InverseProblemPINN<T>

Namespace
AiDotNet.PhysicsInformed.PINNs
Assembly
AiDotNet.dll

Implements a Physics-Informed Neural Network for inverse problems (parameter identification).

public class InverseProblemPINN<T> : NeuralNetworkBase<T>, INeuralNetworkModel<T>, INeuralNetwork<T>, IFullModel<T, Tensor<T>, Tensor<T>>, IModel<Tensor<T>, Tensor<T>, ModelMetadata<T>>, IModelSerializer, ICheckpointableModel, IParameterizable<T, Tensor<T>, Tensor<T>>, IFeatureAware, IFeatureImportance<T>, ICloneable<IFullModel<T, Tensor<T>, Tensor<T>>>, IGradientComputable<T, Tensor<T>, Tensor<T>>, IJitCompilable<T>, IInterpretableModel<T>, IInputGradientComputable<T>, IDisposable

Type Parameters

T

The numeric type used for calculations.

Inheritance
InverseProblemPINN<T>
Implements
IFullModel<T, Tensor<T>, Tensor<T>>
IModel<Tensor<T>, Tensor<T>, ModelMetadata<T>>
IParameterizable<T, Tensor<T>, Tensor<T>>
ICloneable<IFullModel<T, Tensor<T>, Tensor<T>>>
IGradientComputable<T, Tensor<T>, Tensor<T>>
Inherited Members
Extension Methods

Remarks

For Beginners: Inverse problems are about discovering unknown parameters from observations. This is the opposite of forward problems where parameters are known.

Examples of Inverse Problems:

  1. Medical Imaging: Find tumor location from external measurements
  2. Material Science: Identify Young's modulus from stress-strain data
  3. Geophysics: Determine subsurface properties from seismic data
  4. Finance: Calibrate model parameters from market prices

How InverseProblemPINN Works:

  1. Neural network learns the solution u(x,t)
  2. Additional trainable variables represent unknown parameters θ
  3. Both are trained together to minimize:
    • Data loss: ||u_predicted - u_observed||²
    • Physics loss: ||PDE_residual(u, θ)||²
    • Regularization: Prior knowledge about parameters

Key Advantages:

  • Handles noisy and sparse observations
  • Physics acts as regularization
  • No need for iterative PDE solves
  • Can quantify parameter uncertainty

Training Strategy:

  1. Initialize parameters near prior estimates
  2. Train with emphasis on physics initially
  3. Gradually increase data weight
  4. Use separate learning rates for network and parameters

Constructors

InverseProblemPINN(NeuralNetworkArchitecture<T>, IInverseProblem<T>, IBoundaryCondition<T>[], IInitialCondition<T>?, int, InverseProblemOptions<T>?, IGradientBasedOptimizer<T, Tensor<T>, Tensor<T>>?)

Initializes a new instance of the InverseProblemPINN.

public InverseProblemPINN(NeuralNetworkArchitecture<T> architecture, IInverseProblem<T> inverseProblem, IBoundaryCondition<T>[] boundaryConditions, IInitialCondition<T>? initialCondition = null, int numCollocationPoints = 10000, InverseProblemOptions<T>? options = null, IGradientBasedOptimizer<T, Tensor<T>, Tensor<T>>? optimizer = null)

Parameters

architecture NeuralNetworkArchitecture<T>

The neural network architecture.

inverseProblem IInverseProblem<T>

The inverse problem specification.

boundaryConditions IBoundaryCondition<T>[]

Boundary conditions for the problem.

initialCondition IInitialCondition<T>

Initial condition for time-dependent problems (optional).

numCollocationPoints int

Number of collocation points for PDE enforcement.

options InverseProblemOptions<T>

Configuration options for inverse problem training.

optimizer IGradientBasedOptimizer<T, Tensor<T>, Tensor<T>>

Optimization algorithm.

Remarks

For Beginners:

The inverse problem specification defines:

  • What parameters are unknown (e.g., diffusion coefficient)
  • Initial guesses and bounds for these parameters
  • The observations used to identify parameters

The training balances:

  • Fitting observations (data loss)
  • Satisfying physics (PDE residual)
  • Staying near prior estimates (regularization)

Properties

ParameterCount

Gets the total number of parameters in the model.

public override int ParameterCount { get; }

Property Value

int

Remarks

For Beginners: This tells you how many adjustable values (weights and biases) your neural network has. More complex networks typically have more parameters and can learn more complex patterns, but also require more data to train effectively. This is part of the IFullModel interface for consistency with other model types.

Performance: This property uses caching to avoid recomputing the sum on every access. The cache is invalidated when layers are modified.

ParameterHistory

Gets the parameter estimation history.

public IReadOnlyList<T[]> ParameterHistory { get; }

Property Value

IReadOnlyList<T[]>

ParameterNames

Gets the parameter names.

public string[] ParameterNames { get; }

Property Value

string[]

Parameters

Gets the current estimated parameter values.

public T[] Parameters { get; }

Property Value

T[]

SupportsJitCompilation

Gets whether this model currently supports JIT compilation.

public override bool SupportsJitCompilation { get; }

Property Value

bool

True if the model can be JIT compiled, false otherwise.

Remarks

Some models may not support JIT compilation due to: - Dynamic graph structure (changes based on input) - Lack of computation graph representation - Use of operations not yet supported by the JIT compiler

For Beginners: This tells you whether this specific model can benefit from JIT compilation.

Models return false if they:

  • Use layer-based architecture without graph export (e.g., current neural networks)
  • Have control flow that changes based on input data
  • Use operations the JIT compiler doesn't understand yet

In these cases, the model will still work normally, just without JIT acceleration.

SupportsTraining

Indicates whether this network supports training (learning from data).

public override bool SupportsTraining { get; }

Property Value

bool

Remarks

For Beginners: Not all neural networks can learn. Some are designed only for making predictions with pre-set parameters. This property tells you if the network can learn from data.

Methods

CreateNewInstance()

Creates a new instance of the same type as this neural network.

protected override IFullModel<T, Tensor<T>, Tensor<T>> CreateNewInstance()

Returns

IFullModel<T, Tensor<T>, Tensor<T>>

A new instance of the same neural network type.

Remarks

For Beginners: This creates a blank version of the same type of neural network.

It's used internally by methods like DeepCopy and Clone to create the right type of network before copying the data into it.

DeserializeNetworkSpecificData(BinaryReader)

Deserializes network-specific data that was not covered by the general deserialization process.

protected override void DeserializeNetworkSpecificData(BinaryReader reader)

Parameters

reader BinaryReader

The BinaryReader to read the data from.

Remarks

This method is called at the end of the general deserialization process to allow derived classes to read any additional data specific to their implementation.

For Beginners: Continuing the suitcase analogy, this is like unpacking that special compartment. After the main deserialization method has unpacked the common items (layers, parameters), this method allows each specific type of neural network to unpack its own unique items that were stored during serialization.

ForwardWithMemory(Tensor<T>)

Performs a forward pass through the network while storing intermediate values for backpropagation.

public override Tensor<T> ForwardWithMemory(Tensor<T> input)

Parameters

input Tensor<T>

The input data to the network.

Returns

Tensor<T>

The output of the network.

Remarks

For Beginners: This method passes data through the network from input to output, but also remembers all the intermediate values. This is necessary for the learning process, as the network needs to know these values when figuring out how to improve.

API Change Note: The signature changed from Vector<T> to Tensor<T> to support multi-dimensional inputs. This is a breaking change. For backward compatibility, consider adding an overload that accepts Vector<T> and converts it internally to Tensor<T>.

Exceptions

InvalidOperationException

Thrown when the network doesn't support training.

GetGradients()

Gets the gradients from all layers in the neural network.

public override Vector<T> GetGradients()

Returns

Vector<T>

A vector containing all gradients from all layers concatenated together.

Remarks

This method collects the gradients from every layer in the network and combines them into a single vector. This is useful for optimization algorithms that need access to all gradients at once.

For Beginners: During training, each layer calculates how its parameters should change (the gradients). This method gathers all those gradients from every layer and puts them into one long list.

Think of it like:

  • Each layer has notes about how to improve (gradients)
  • This method collects all those notes into one document
  • The optimizer can then use this document to update the entire network

This is essential for the learning process, as it tells the optimizer how to adjust all the network's parameters to improve performance.

GetModelMetadata()

Gets the metadata for this neural network model.

public override ModelMetadata<T> GetModelMetadata()

Returns

ModelMetadata<T>

A ModelMetaData object containing information about the model.

GetParameters()

Gets all trainable parameters of the network as a single vector.

public override Vector<T> GetParameters()

Returns

Vector<T>

A vector containing all parameters of the network.

Remarks

For Beginners: Neural networks learn by adjusting their "parameters" (also called weights and biases). This method collects all those adjustable values into a single list so they can be updated during training.

InitializeLayers()

Initializes the layers of the neural network based on the architecture.

protected override void InitializeLayers()

Remarks

For Beginners: This method sets up all the layers in your neural network according to the architecture you've defined. It's like assembling the parts of your network before you can use it.

Predict(Tensor<T>)

Makes a prediction using the neural network.

public override Tensor<T> Predict(Tensor<T> input)

Parameters

input Tensor<T>

The input data to process.

Returns

Tensor<T>

The network's prediction.

Remarks

For Beginners: This is the main method you'll use to get results from your trained neural network. You provide some input data (like an image or text), and the network processes it through all its layers to produce an output (like a classification or prediction).

SerializeNetworkSpecificData(BinaryWriter)

Serializes network-specific data that is not covered by the general serialization process.

protected override void SerializeNetworkSpecificData(BinaryWriter writer)

Parameters

writer BinaryWriter

The BinaryWriter to write the data to.

Remarks

This method is called at the end of the general serialization process to allow derived classes to write any additional data specific to their implementation.

For Beginners: Think of this as packing a special compartment in your suitcase. While the main serialization method packs the common items (layers, parameters), this method allows each specific type of neural network to pack its own unique items that other networks might not have.

Solve(int, double, bool)

Solves the inverse problem to identify unknown parameters.

public InverseProblemResult<T> Solve(int epochs = 1000, double learningRate = 0.001, bool verbose = true)

Parameters

epochs int

Number of training epochs.

learningRate double

Learning rate for the neural network.

verbose bool

Whether to print progress.

Returns

InverseProblemResult<T>

Results including identified parameters and uncertainties.

Train(Tensor<T>, Tensor<T>)

Trains the neural network on a single input-output pair.

public override void Train(Tensor<T> input, Tensor<T> expectedOutput)

Parameters

input Tensor<T>

The input data.

expectedOutput Tensor<T>

The expected output for the given input.

Remarks

This method performs one training step on the neural network using the provided input and expected output. It updates the network's parameters to reduce the error between the network's prediction and the expected output.

For Beginners: This is how your neural network learns. You provide: - An input (what the network should process) - The expected output (what the correct answer should be)

The network then:

  1. Makes a prediction based on the input
  2. Compares its prediction to the expected output
  3. Calculates how wrong it was (the loss)
  4. Adjusts its internal values to do better next time

After training, you can get the loss value using the GetLastLoss() method to see how well the network is learning.

UpdateParameters(Vector<T>)

Updates the network's parameters with new values.

public override void UpdateParameters(Vector<T> parameters)

Parameters

parameters Vector<T>

The new parameter values to set.

Remarks

For Beginners: During training, a neural network's internal values (parameters) get adjusted to improve its performance. This method allows you to update all those values at once by providing a complete set of new parameters.

This is typically used by optimization algorithms that calculate better parameter values based on training data.

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