Dual-Level Models for Physics-Informed Multi-Step Time Series Forecasting

• URL: http://arxiv.org/abs/2601.07640v1
• Date: Mon, 12 Jan 2026 15:19:21 GMT
• Title: Dual-Level Models for Physics-Informed Multi-Step Time Series Forecasting
• Authors: Mahdi Nasiri, Johanna Kortelainen, Simo Särkkä,
• Abstract summary: This paper develops an approach for multi-step forecasting of dynamical systems by integrating probabilistic input forecasting with physics-informed output prediction.<n>It shows that the hybrid input forecasting models achieve a higher log-likelihood and lower mean squared errors compared to conventional STMs.
• Score: 7.227766714611374
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: This paper develops an approach for multi-step forecasting of dynamical systems by integrating probabilistic input forecasting with physics-informed output prediction. Accurate multi-step forecasting of time series systems is important for the automatic control and optimization of physical processes, enabling more precise decision-making. While mechanistic-based and data-driven machine learning (ML) approaches have been employed for time series forecasting, they face significant limitations. Incomplete knowledge of process mathematical models limits mechanistic-based direct employment, while purely data-driven ML models struggle with dynamic environments, leading to poor generalization. To address these limitations, this paper proposes a dual-level strategy for physics-informed forecasting of dynamical systems. On the first level, input variables are forecast using a hybrid method that integrates a long short-term memory (LSTM) network into probabilistic state transition models (STMs). On the second level, these stochastically predicted inputs are sequentially fed into a physics-informed neural network (PINN) to generate multi-step output predictions. The experimental results of the paper demonstrate that the hybrid input forecasting models achieve a higher log-likelihood and lower mean squared errors (MSE) compared to conventional STMs. Furthermore, the PINNs driven by the input forecasting models outperform their purely data-driven counterparts in terms of MSE and log-likelihood, exhibiting stronger generalization and forecasting performance across multiple test cases.

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