Technical Summary: Time-Input State-Aware Calibration (TISAC)

1. Problem Statement

Physics-based modeling of engineering systems often relies on governing equations linking state variables (operating conditions) and uncertain input parameters to system responses. However, discrepancies between model predictions and reality arise from uncertain parameters, parameter misspecification, and incomplete physics representations.

Traditional model calibration frameworks, such as the seminal Kennedy and O'Hagan (KOH) approach, typically assume calibration parameters are global constants or vary only with time or state independently. This assumption is insufficient for many modern engineering systems (e.g., degrading batteries, aging aero-engines, or grid-connected inverters) where parameters evolve jointly with system state and time. When parameters are treated as static or purely time-dependent while they are actually state-dependent, the calibration process suffers from confounding: the model cannot distinguish between errors caused by incorrect parameter values and errors caused by structural model deficiencies (discrepancy). This leads to biased parameter estimates and poor predictive performance under changing operational conditions.

Furthermore, existing methods often rely on Gaussian Processes (GPs) for emulation and discrepancy modeling. While analytically tractable, GPs face computational bottlenecks (cubic time complexity, $O(n^3)$) with high-dimensional or extensive time-series data, often necessitating dimensionality reduction that assumes stationarity.

2. Methodology: TISAC Framework

The authors propose Time-Input State-Aware Calibration (TISAC), a framework that explicitly models calibration parameters as functions of both system state $x$ and time $t$, while simultaneously inferring a time- and state-dependent discrepancy field.

2.1 Mathematical Formulation

The observation model is formulated as:
$$y_{obs}(t, x) = \eta(t, x, \theta_t(x, t), \theta_s(t)) + \delta(x, t) + \epsilon(t)$$
Where:

• $\eta$ is the physics-based model.
• $\theta_t(x, t)$ represents parameters evolving with both state and time.
• $\theta_s(t)$ represents parameters evolving only with time.
• $\delta(x, t)$ is the model discrepancy field.
• $\epsilon(t)$ is experimental noise.

2.2 Co-Evolving Neural Network Architecture

TISAC employs a unified architecture of three feed-forward neural networks that co-evolve during training:

1. Parameter Networks ($NN_{\theta_t}, NN_{\theta_s}$): These infer the calibration fields. They are structured to output bounded point estimates corrected from structural priors using a hyperbolic tangent activation to confine residuals.
2. Discrepancy Network ($NN_{\delta}$): This infers the discrepancy field $\delta(x, t)$, regularized to capture smooth, systematic model-form errors rather than random noise.
3. Response Network (Optional): In cases where an explicit differentiable governing equation is unavailable (e.g., the power system case), a surrogate network maps inputs and inferred parameters to outputs. In synthetic benchmarks, the response is obtained by integrating the governing ODE using a differentiable forward-Euler scheme.

2.3 Loss Function and Regularization

The framework minimizes a joint objective loss function ($L_{total}$) composed of several terms:

• Data-fitting Loss ($L_y$): Measures the difference between predicted and observed responses, scaled by empirical standard deviations.
• Structural Priors ($L_{\theta, prior}$): Encodes expected dependency classes and smooth trends for parameters.
• Smoothness and Range Penalties ($L_{\theta, smooth}, L_{\theta, range}$): Enforces physical admissibility and smoothness in space and time.
• Discrepancy Regularization ($L_{\delta}$): Includes energy, smoothness (first and second derivatives), edge (boundary zeroing), and spectral penalties. The spectral penalty suppresses high-frequency components in the discrepancy to prevent it from absorbing measurement noise or overfitting.
• Separation Loss ($L_{sep}$): A critical innovation that penalizes the alignment between the discrepancy vector and the sensitivity vectors of the calibration parameters. This encourages the discrepancy to correct for structural errors in directions orthogonal to parameter variations, thereby reducing confounding.

2.4 Optimization and Encoding

To address the spectral bias of feed-forward neural networks (which struggle to learn high-frequency variations), the inputs are mapped using Fourier positional encoding. The optimization uses stochastic gradient descent (Adam) with mini-batching, a warmup phase, and a cosine learning rate decay schedule to ensure stable convergence.

3. Key Contributions

1. Joint State-Time Calibration: The framework explicitly disentangles parameters that evolve jointly with state and time, addressing a gap in literature where time-dependence and state-dependence were treated separately.
2. Reduced Confounding: By introducing a separation loss based on ODE sensitivity vectors and enforcing structural priors, TISAC significantly reduces the identifiability issues between calibration parameters and model discrepancy.
3. Scalable Neural Surrogates: Replacing Gaussian Processes with neural networks allows for efficient handling of high-dimensional, time-series data without the cubic computational cost of covariance matrix inversion.
4. Physics-Informed Constraints: The method integrates governing physics (via ODE integration or sensitivity analysis) directly into the loss function, ensuring physically meaningful parameter inference.

4. Results

4.1 Academic Case Study (Synthetic Benchmark)

The framework was tested on a synthetic ODE system with known ground truth for parameters and discrepancy.

• Performance: TISAC successfully recovered the true state- and time-varying calibration fields and discrepancy.
• Comparison: Compared to a standard KOH-style Bayesian calibration (using GP emulators and constant parameters), TISAC drastically reduced prediction errors. For the test case (Measurement 3), the Root Mean Square Error (RMSE) for the response dropped from 63.83% (Bayesian) to 3.21% (TISAC).
• Parameter Inference: The error in inferring the calibration parameter decreased from 22.31% to 1.83%, and discrepancy inference error dropped from 33.63% to 14.48%, demonstrating the effectiveness of the separation mechanism.
• Data Scenarios: The framework remained robust across different data sampling densities (state-rich vs. time-rich) and sparse data gaps.

4.2 Power System Application

The method was applied to a high-fidelity model of a grid-forming inverter in an integrated transmission and distribution system.

• Setup: The model inferred grid impedance (time-dependent) and grid inertia (state- and time-dependent) under varying load conditions.
• Results:
  • Training Region: Normalized RMSE for power, voltage, and frequency decreased from ~96-108% (pre-calibration) to 1.4%, 1.1%, and 0.9% respectively.
  • Holdout Region: The model maintained accuracy in unseen operating segments and future time steps, with errors reducing to 10.9% (power), 9% (voltage), and 9.5% (frequency).
  • Discrepancy: The inferred discrepancy correctly captured the direction of model-form errors in the holdout region, providing necessary corrections to improve response predictions.

5. Significance and Claims

The paper claims that TISAC offers a physically grounded approach for the performance monitoring and operational management of dynamic engineering systems. Its primary significance lies in its ability to:

• Support more reliable decision-making under changing real-world operating conditions by accurately separating parameter uncertainty from model structural error.
• Provide a scalable alternative to Gaussian Process-based calibration for complex, time-evolving systems.
• Enable the development of "physically grounded digital twin" workflows for systems where parameters are not static but co-evolve with system states.

The authors note that the methodology is currently most applicable to systems with stable dynamics and smooth residual behavior, with future work needed to address rapid, discontinuous responses. The framework is presented as a step toward more robust calibration for complex domains such as aerospace propulsion, smart manufacturing, and battery monitoring.