Progressive multi-fidelity learning for physical system predictions

TL;DR

The paper presents a progressive multi-fidelity surrogate that sequentially fuses heterogeneous, multimodal data via per-level encoders and additive output corrections to predict high-fidelity physical quantities. Training proceeds offline level by level with latent fusion and a penalty on overfitting, while online deployment adapts to the available low-fidelity inputs to maintain continuous, accurate predictions. Across reaction–diffusion, Navier–Stokes, and air-pollution benchmarks, the approach yields systematic accuracy gains at each level, with substantial online speed-ups and ensemble-based uncertainty quantification that reveals reliability even under missing HF data. The framework demonstrates robust generalization across time and parameter variations, offering practical benefits for real-time scientific computing in data-scarce regimes and multi-modal settings.

Abstract

Highly accurate datasets from numerical or physical experiments are often expensive and time-consuming to acquire, posing a significant challenge for applications that require precise evaluations, potentially across multiple scenarios and in real-time. Even building sufficiently accurate surrogate models can be extremely challenging with limited high-fidelity data. Conversely, less expensive, low-fidelity data can be computed more easily and encompass a broader range of scenarios. By leveraging multi-fidelity information, prediction capabilities of surrogates can be improved. However, in practical situations, data may be different in types, come from sources of different modalities, and not be concurrently available, further complicating the modeling process. To address these challenges, we introduce a progressive multi-fidelity surrogate model. This model can sequentially incorporate diverse data types using tailored encoders. Multi-fidelity regression from the encoded inputs to the target quantities of interest is then performed using neural networks. Input information progressively flows from lower to higher fidelity levels through two sets of connections: concatenations among all the encoded inputs, and additive connections among the final outputs. This dual connection system enables the model to exploit correlations among different datasets while ensuring that each level makes an additive correction to the previous level without altering it. This approach prevents performance degradation as new input data are integrated into the model and automatically adapts predictions based on the available inputs. We demonstrate the effectiveness of the approach on numerical benchmarks and a real-world case study, showing that it reliably integrates multi-modal data and provides accurate predictions, maintaining performance when generalizing across time and parameter variations.

Figures (8)

• Figure 1: Progressive multi-fidelity surrogate modeling structure. The method is composed of three steps: (i) input encoding, at each level $l$, input data ${\mathbf{x}}^{(l)}$ are mapped by encoders $\Phi^{(l)}$ into latent variables ${\mathbf{h}}^{(l)}$; (ii) latent concatenation and decoding, latent variables are concatenated and decoded by $\Psi^{(l)}$; (iii) output summation, previous output is summed to the current one to form the final output ${\hat{\mathbf{y}}}^{(l)}$.
• Figure 2: Multi-fidelity model for generating high-fidelity solutions to a reaction-diffusion system. At level 0 a deep-learning surrogate is constructed to estimate spatio-temporal solutions from the control parameters (time and reaction coefficient). In subsequent levels, information from artificial boundary sensors and cheap, low-resolution solutions are progressively integrated into the model. Encoder networks $\{\Phi^{(i)}\}_{i=0}^2$ process data from different modalities in a compatible latent representations. The decoder networks $\{\Psi^{(i)}\}_{i=0}^2$ at each level reconstruct spatio-temporal solutions from the latent representations by means of LSTM networks and POD.
• Figure 3: Progressive multi-fidelity prediction (left) for each fidelity level and absolute error (right) with respect to the high-fidelity reference solutions of the reaction-diffusion problem. The illustrated snapshots refer to one interpolated time instance $t=20$ and two extrapolated time instances $t\in\{50,70\}$ (being $T_\text{train}=40$ the end of the HF training coverage) for two testing values of reaction parameter $\mu\in\{0.875,1.375\}$ (top and bottom plots, respectively), unseen during the training.
• Figure 4: Comparison of the time evolution of the expansion coefficients of the POD modes (POM) in the reaction-diffusion example between the HF reference and the MF predicted solution. For the MF predictions, mean $\hat{y}$ and two standard deviation $\sigma$ band over the different trainings are shown, providing uncertainty quantification. The vertical dashed line at $T_\text{train}= 40$ indicates the end time of HF data coverage in training, i.e., no HF information is available for $t\in [T_\text{train}, T].$
• Figure 5: Multi-fidelity model for generating high-fidelity solutions to the Navier-Stokes equations. At level 0 a deep-learning surrogate is constructed to estimate spatio-temporal solutions from the input parameters (time and Reynolds number coefficient). At level 1 information about the time evolution of drag and lift coefficients, expressing the forces acting on the obstacle wall, are integrated. At level 2 information about solution outflow, representing boundary sensors, is included. Finally, spatio-temporal snapshots of a limited portion of the domain (surrounding the obstacle) are assimilated by the model at level 3. Encoder networks $\{\Phi^{(i)}\}_{i=0}^3$ process data from different modalities into compatible latent representations. The decoder networks $\{\Psi^{(i)}\}_{i=0}^3$ at each level reconstruct spatio-temporal solutions from the latent representation by means of LSTM networks and POD.