Bridging Structured Knowledge and Data: A Unified Framework with Finance Applications

Abstract

We develop Structured-Knowledge-Informed Neural Networks (SKINNs), a unified estimation framework that embeds theoretical, simulated, previously learned, or cross-domain insights as differentiable constraints within flexible neural function approximation. SKINNs jointly estimate neural network parameters and economically meaningful structural parameters in a single optimization problem, enforcing theoretical consistency not only on observed data but over a broader input domain through collocation, and therefore nesting approaches such as functional GMM, Bayesian updating, transfer learning, PINNs, and surrogate modeling. SKINNs define a class of M-estimators that are consistent and asymptotically normal with root-N convergence, sandwich covariance, and recovery of pseudo-true parameters under misspecification. We establish identification of structural parameters under joint flexibility, derive generalization and target-risk bounds under distributional shift in a convex proxy, and provide a restricted-optimal characterization of the weighting parameter that governs the bias-variance tradeoff. In an illustrative financial application to option pricing, SKINNs improve out-of-sample valuation and hedging performance, particularly at longer horizons and during high-volatility regimes, while recovering economically interpretable structural parameters with improved stability relative to conventional calibration. More broadly, SKINNs provide a general econometric framework for combining model-based reasoning with high-dimensional, data-driven estimation.

Figures (14)

• Figure 1: PSKR by a (semi-)parametric equation. $\mathbf{x}^{\text{SK}}_{1}\cdots\mathbf{x}^{\text{SK}}_{d_{\text{SK}}}$ denote each of the observable features, and $\mathbf{\phi}_{1}\cdots\mathbf{\phi}_{d_{\mathbf{\phi}}}$ denote each of the learnable latent parameters, in the PSKR.
• Figure 2: SPSKR by an expensive-to-solve target system. The inputs, i.e., the observable features $\mathbf{X}^{\text{SK}}$ and the latent parameters $\mathbf{\phi}$, are still determined by theories, as in PSKRs. But the structured-knowledge function $g_{\mathbf{\phi}}(\mathbf{X}^{\text{SK}})$, in this case, is approximated by a pre-trained DSNN.
• Figure 3: NPSKR by an unknown distribution. The inputs include a high-dimensional vector of probabilities, which serves as the learnable latent parameters in SKINNs. The probabilities are passed to a softmax function to ensure non-negativity and the total probability of one unit.
• Figure 4: NPSKR by an AE. The inputs include a vector of realizations from an unknown DGP, with size $n$. The encoder compresses the inputs into a vector of latent parameters of size $d$ ($d\ll n$). The $d$-dimensional AE latent parameters are then passed to a decoder, which returns an $n$-dimensional vector that reconstructs the inputs.
• Figure 5: The architecture of SKINNs. The NN on the top (blue circles) is the base function approximator. The mappings at the bottom (red circles) are structured-knowledge representations of different formats. All of them take observable features $\mathbf{X}^{\text{SK}}_{\text{Colloc}}$; the current epoch value of the learnable latent parameters as inputs, and output $g_{\phi}$.

Theorems & Definitions (20)

• Lemma 1: Identification through profiling
• proof
• Proposition 1: A sufficient condition
• proof
• Theorem 1: Consistency
• proof
• Theorem 2: Asymptotic Normality
• proof
• Proposition 2: Restricted-optimal $\lambda$
• proof