Shivansh

Bridging Mathematics and AI to Model the Physical World.

• The Challenge: How do you force a neural network to obey the laws of physics?
• The Solution: By minimizing a composite loss function that simultaneously enforces known data points and the residual of the governing Partial Differential Equation ($\mathcal{F}(u)$).

$$\mathcal{L}(\theta) = \underbrace{\lambda_{data} \mathcal{L}_{data}}_{\text{Data Fidelity}} + \underbrace{\lambda_{physics} \mathcal{L}_{physics}}_{\text{Physical Law Constraint}}$$

• The Challenge: How can a network represent continuous, complex signals without losing detail?
• The Solution: By constructing a network where each layer is a sinusoidal function, allowing it to perfectly capture the signal's derivatives and fine details.

$$\Phi(\mathbf{x}) = (f_L \circ \dots \circ f_1)(\mathbf{x}) \quad \text{where} \quad f_i(\mathbf{x}_i) = \sin(\mathbf{W}_i \mathbf{x}_i + \mathbf{b}_i)$$