Mathematical Framework of NeuralFlow

NeuralFlow is formulated as an implicit optimization problem where a neural network is trained to minimize the residuals of a governing system of partial differential equations (PDEs). Given a set of state variables \( V \), the solver constructs a numerical representation of fluxes \( F \), ensuring conservation laws and physical consistency through automatic differentiation.

Governing Equations

The state variables \( V \) satisfy a general system of governing equations expressed as:

\( \mathcal{R}(V) = 0 \)

where \( \mathcal{R} \) represents the residuals derived from conservation laws:

\( \frac{\partial V}{\partial t} + \nabla \cdot F(V) = S(V) \)

where:

• \( F(V) \) is the flux function.
• \( S(V) \) represents source terms.

Neural Network Approximation

NeuralFlow approximates the state variables using a neural network \( \mathcal{N}_\theta \):

\( V = \mathcal{N}_\theta(X) \)

where \( X \) represents input features such as spatial coordinates, initial conditions, or auxiliary parameters. The objective is to train the network to minimize the residuals \( \mathcal{R}(V) \).

Flux Jacobians and Linearization

Using automatic differentiation, the Jacobian of the flux function is computed as:

\( J_F = \frac{\partial F}{\partial V} \)

allowing for a local linearization:

\( \delta F \approx J_F \delta V \)

Optimization Problem

The optimization objective is to minimize the total residual loss over the computational domain:

\( \mathcal{J}(\theta) = \sum_{\Omega} \|\mathcal{R}(V)\|^2 \)

The network is trained using a gradient-based optimizer, where gradients are corrected using the flux Jacobians to enforce physics consistency:

\( \frac{\partial \mathcal{J}}{\partial \theta} = \sum_{i} \frac{\partial \mathcal{J}}{\partial V_i} \frac{\partial V_i}{\partial \theta} \)

Final Update Rule

The parameter updates are performed iteratively:

\( \theta \leftarrow \theta - \eta \frac{\partial \mathcal{J}}{\partial \theta} \)

where \( \eta \) is the learning rate.

Conclusion

By coupling numerical discretization with differentiable programming, NeuralFlow ensures that the learned state variables adhere to the governing physics while enabling a data-driven approach to solving PDEs.