Table of Contents

Class UniversalDifferentialEquation<T>

Namespace
AiDotNet.PhysicsInformed.ScientificML
Assembly
AiDotNet.dll

Implements Universal Differential Equations (UDEs) - ODEs with neural network components.

public class UniversalDifferentialEquation<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
UniversalDifferentialEquation<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: Universal Differential Equations combine known physics with machine learning.

Traditional ODEs: dx/dt = f(x, t, θ) where f is a known function with parameters θ Example: dx/dt = -kx (exponential decay, k is known)

Pure Neural ODEs: dx/dt = NN(x, t, θ) where NN is a neural network

  • Very flexible, can learn any dynamics
  • But ignores known physics
  • May violate physical laws

Universal Differential Equations (UDEs): dx/dt = f_known(x, t) + NN(x, t, θ)

  • Combines known physics (f_known) with learned corrections (NN)
  • Best of both worlds!

Key Idea: Use neural networks to model UNKNOWN parts of the physics while keeping KNOWN parts as explicit equations.

Example - Epidemic Model: Known: dS/dt = -βSI, dI/dt = βSI - γI (basic SIR model) Unknown: How β (infection rate) varies with temperature, policy, etc. UDE: dS/dt = -β(T, P)SI where β(T, P) = NN(temperature, policy)

Applications:

  • Climate modeling (known physics + unknown feedback loops)
  • Epidemiology (known disease spread + unknown interventions)
  • Chemistry (known reactions + unknown catalysis effects)
  • Biology (known population dynamics + unknown environmental factors)
  • Engineering (known mechanics + unknown friction/damping)

Constructors

UniversalDifferentialEquation(NeuralNetworkArchitecture<T>, int, Func<T[], T, T[]>?, IGradientBasedOptimizer<T, Tensor<T>, Tensor<T>>?) 

public UniversalDifferentialEquation(NeuralNetworkArchitecture<T> architecture, int stateDim, Func<T[], T, T[]>? knownDynamics = null, IGradientBasedOptimizer<T, Tensor<T>, Tensor<T>>? optimizer = null)

Parameters

architecture NeuralNetworkArchitecture<T>
stateDim int
knownDynamics Func<T[], T, T[]>
optimizer IGradientBasedOptimizer<T, Tensor<T>, Tensor<T>>

Properties

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 model supports training.

public override bool SupportsTraining { get; }

Property Value

bool

Methods

Backward(Tensor<T>)

Performs a backward pass through the network (backpropagation).

public Tensor<T> Backward(Tensor<T> outputGradient)

Parameters

outputGradient Tensor<T>

Gradient of the loss with respect to network output.

Returns

Tensor<T>

Gradient of the loss with respect to input.

ComputeDerivative(T[], T)

Computes dx/dt = f_known(x, t) + NN(x, t).

public T[] ComputeDerivative(T[] state, T time)

Parameters

state T[]
time T

Returns

T[]

CreateNewInstance()

Creates a new instance with the same configuration.

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

Returns

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

New UDE instance.

DeserializeNetworkSpecificData(BinaryReader)

Deserializes UDE-specific data.

protected override void DeserializeNetworkSpecificData(BinaryReader reader)

Parameters

reader BinaryReader

Binary reader.

Forward(Tensor<T>)

Performs a forward pass through the network.

public Tensor<T> Forward(Tensor<T> input)

Parameters

input Tensor<T>

Input tensor for evaluation.

Returns

Tensor<T>

Network output tensor.

GetModelMetadata()

Gets metadata about the UDE model.

public override ModelMetadata<T> GetModelMetadata()

Returns

ModelMetadata<T>

Model metadata.

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 UDE model.

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

Parameters

input Tensor<T>

Input tensor with state and time.

Returns

Tensor<T>

Predicted derivative tensor.

SerializeNetworkSpecificData(BinaryWriter)

Serializes UDE-specific data.

protected override void SerializeNetworkSpecificData(BinaryWriter writer)

Parameters

writer BinaryWriter

Binary writer.

Simulate(T[], T, T, int, OdeIntegrationMethod)

Simulates the UDE forward in time.

public T[,] Simulate(T[] initialState, T tStart, T tEnd, int numSteps, OdeIntegrationMethod method = OdeIntegrationMethod.Euler)

Parameters

initialState T[]
tStart T
tEnd T
numSteps int
method OdeIntegrationMethod

Returns

T[,]

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

Performs a supervised training step against derivative targets.

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

Parameters

input Tensor<T>

Input tensor with state and time.

expectedOutput Tensor<T>

Expected derivative tensor.

Remarks

Uses standard backpropagation like all other neural networks. The network learns to predict the derivative (dx/dt) at each state.

UpdateParameters(Vector<T>)

Updates the network parameters from a flattened vector.

public override void UpdateParameters(Vector<T> parameters)

Parameters

parameters Vector<T>

Parameter vector.

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