Physics-Informed Forecasting

This work develops a physics-informed multimodal deep learning framework for 10- to 110-minute-ahead probabilistic photovoltaic power forecasting. The forecasting model is built on a four-branch architecture that separately encodes plant-level temporal history and inverter-level spatial cloud dynamics through a graph attention encoder with a learned adjacency, known-future solar geometry, and the current meteorological state, fused through gated cross-attention. The geometry branch adds two forward-looking physical signals, spatial cloud heterogeneity and inverter ramp-down activity, each empirically weighted toward the horizon where it is most predictive. The decoder anchors its prediction to a smart persistence baseline, learning a horizon-conditioned correction, and the resulting quantile forecasts are calibrated online through a kernel-weighted adaptive conformal calibration step. The proposed model outperforms all 14 baselines, achieving a forecast skill score (FSS) of 0.2950, 0.099 above the next-best model. A branch-composition ablation shows that every branch contributes positively and the model retains substantial skill in challenging overcast and variable sky conditions (FSS 0.198 and 0.255).

Physics-informed machine learning (PIML) and hybrid modeling have attracted significant attention as approaches for combining physical knowledge with the flexibility of deep learning. In this work, we investigate whether these approaches deliver their commonly claimed benefits for large-scale photovoltaic (PV) power forecasting. We develop two hybrid forecasting architectures that integrate the ADR physical PV efficiency model with the TSMixer neural forecasting architecture through feature augmentation and residual learning. The proposed models are evaluated on a dataset comprising over 3 million samples from 3,713 PV installations across 33 European countries. We assess convergence speed, forecasting accuracy, transferability across geographical regions, and data efficiency. Although the physics-informed models seem to exhibit slightly faster convergence during training, they do not achieve consistent improvements in test-set accuracy, generalization, or performance under limited training data compared to the purely data-driven baseline. These findings suggest that, for modern deep forecasting architectures trained on large datasets, explicitly incorporating the ADR physical model provides limited practical benefit for day-ahead PV forecasting.

Advection-diffusion plays an essential role in transport phenomena such as atmospheric and cloud dynamics. Improved forecasting of cloud motion, in turn, enhances solar power production forecasts and facilitates the integration of additional photovoltaic (PV) sources into the power grid. However, both classical numerical models and emerging deep learning approaches face challenges when applied to spatially irregular sensor networks such as those formed by weather stations. In this work, we adopt a graph signal processing perspective and model advection-diffusion particles as signals on a graph to capture the dynamics of these processes. We introduce a Physics-Informed Graph Neural Network (PING) for forecasting future advection-diffusion particle concentrations on irregular grids. The model incorporates a discretized advection-diffusion equation as a soft constraint in the loss function to guide learning toward physically consistent solutions. This model is on par with state of the art models on synthetic and real transport phenomena datasets, including sea surface temperature and cloud index datasets. Furthermore, the PING model is successfully coupled with models for forecasting PV power production. This hybrid approach leverages data fusion from ground-based sensors at PV stations and satellite images with cloud concentration to improve the accuracy of intra-day PV power production.

Spatiotemporal forecasting on hydrologic graphs is especially prone to instability in open-boundary systems, where the forecast domain exchanges fluxes with an unobserved exterior. In such systems, boundary nodes receive external forcing, e.g., upstream inflows in rivers or tidal signals in coastal regions, that is typically unavailable at prediction time. The absence of this information can compound errors as forecasts unfold in an autoregressive fashion, leading to inferior long-horizon performance. This paper dissects this instability issue by exploring two questions. 1) What boundary forcing enters the forecast domain when information beyond the boundary is missing? 2) How should this forcing propagate through the domain without incurring error amplification under autoregressive rollout? To address both, we propose a new computing framework comprising two key components. First, to compensate for the boundary forcing, our framework learns ghost node proxies from the boundary and interior nodes, striving to approximate unobserved external inputs. Second, to control error accumulation from these learned proxies, we leverage two physics refiners. In particular, one refiner enforces local consistency by aligning ghost proxies with their two-hop neighbors (i.e., boundary nodes and their immediate interiors). The other refiner enhances global stability by correcting the model forecasts through a physics-guided graph neural operator, reducing long-horizon numerical drift. Two real-world hydrologic graphs are employed for empirical evaluation. Comparative results show that our proposal enjoys higher prediction accuracy and long-horizon stability over both learning-based and physics-informed model competitors.

Time series forecasting has seen signicant advancements with the emergence of new deep learning models. However, forecasting time series in applications involving physical processes remains a major challenge. Despite the apparition of Physics Informed Neural Networks (PINN), recent models do not estimate unobservable intermediate physical variables, which are important for domain experts to understand the target behavior. To this end, we propose a Physics Informed Recurrent Neural Network (PIRNN) which predicts, along the target, unobservable variables on both historic data and forecast target. This approach enhances the model robustness and results interpretation using domain knowledge. Our method is easily adaptable to any physical model using several equations, each having its own set of unobservable variables, to describe it-self. As a case study, we incorporate physical equations used for groundwater levels predictions by the physical model called Gardenia. This model uses transfers equations between reservoirs, optimized with data assimilation, to simulate the evolution of groundwater levels. Evaluation includes several well known neural network models and the Gardenia model compared on twelve real world datasets. In addition, we study the impact of each component through an ablation study. Our model outperforms other models on ve out of the twelve datasets and our ablation study underlines the importance of having a physical background in our time series forecasting task. Finally, the coherence of the physical variables predicted by our neural network is assessed by a domain expert.

Accurate station-level precipitation nowcasting is critical for agriculture, water resource management, and disaster prevention, which typically is formulated as a time series forecasting problem. However, conventional time-series modeling techniques face two major challenges in addressing station-level precipitation nowcasting: (1) Lack of Physics-Guided Modeling}, where meteorological variables are treated as a homogeneous set without accounting for their distinct roles in precipitation formation, leads to predictions that deviate from the physical processes governing precipitation. (2) Severe zero inflation in precipitation, where dry intervals dominate the dataset, obscuring meaningful precipitation patterns and complicating the predictive modeling. To address these challenges, we propose MZ-Rain, a moisture-budget-guided zero-inflated sLSTM framework for station-level precipitation nowcasting. Guided by the moisture budget equation, MZ-Rain decomposes the precipitation formation process into process-specific pathways corresponding to moisture storage, moisture transport, surface evaporation, and precipitation persistence, and captures their temporal evolution through dedicated sLSTM branches. To account for the zero-inflated nature of precipitation, MZ-Rain introduces an adaptive Tweedie modeling strategy that adaptively modulates the rainfall mean while jointly learning precipitation occurrence as an auxiliary task, enabling the model to better balance dry-wet discrimination and quantitative precipitation estimation. Extensive experiments across diverse geographical and climatic regimes demonstrate that MZ-Rain consistently outperforms strong baselines on multiple evaluation metrics, including CSI, FAR, MSE, and MAE. In particular, the model exhibits superior skill in forecasting heavy precipitation events, while benefiting from physically grounded process modeling.

Deep generative models such as flow matching and diffusion models have shown potential for learning complex dynamical systems, but typically act as black boxes that neglect underlying physical structure, while physics-based models governed by partial differential equations are often incomplete due to missing source terms, or uncertain parametrisations. We present Climate Physics Dynamic Matching (ClimPhyDM), a variational simulation-free dynamics informed framework for weather forecasting that combines an advection-type physics prior with data-driven components in a variational framework. % to capture the stochasticity and multi-modality of unresolved atmospheric dynamics. On the ERA5 benchmark at hourly (42-hour) and monthly (5-month) resolutions, ClimPhyDM outperforms ClimODE, and GB-DM, keeping the lower error at extended horizon, indicating improved temporal stability and resistance to error accumulation, while its simulation-free paradigm also enables training on a single modest 12 GB consumer GPU.

We study a hybrid proposal--correction architecture for reduced-order PDE forecasting in which a pure-state quantum reservoir computer (QRC) predicts latent coefficient dynamics and a PINN-based physics-informed corrector (PIC) refines local rollout windows. The method is evaluated on Burgers and Kuramoto--Sivashinsky (KS), with KS as the primary chaotic benchmark. On KS, QRC+PIC consistently improves over QRC alone in RMSE, NRMSE, and PDE residual, while Burgers highlights a regime in which simple baselines remain strong. These results suggest that QRC proposals with local physics-informed correction are a viable benchmark-dependent reduced-order forecasting strategy.

Understanding object dynamics requires not only predicting future trajectories but also examining whether a model captures the physical properties that govern motion. However, existing benchmarks rarely expose object-level physical properties as explicit evaluation targets alongside trajectory forecasting. To address this gap, we introduce ExPhy, a multi-object trajectory forecasting benchmark containing 24,000 simulated physical scenes with explicit object-level labels for mass, friction, and restitution. ExPhy provides observed and future trajectories together with an in-distribution (ID) split and two out-of-distribution (OOD) splits over physical parameters (OOD-Parameter) and initial states (OOD-Initial) for jointly evaluating trajectory forecasting and physical property estimation. We further instantiate PhyODE, a physics-guided model with an explicit property interface that estimates physical properties from observed trajectories and uses them for differentiable future rollout. On the long-horizon OOD-Initial setting, PhyODE reduces ADE and FDE by 33.1% and 31.0%, respectively, compared with the strongest baseline. Zero-shot evaluation on ComPhy further assesses cross-benchmark transfer. Property-level analyses reveal that accurate trajectory forecasting does not necessarily imply accurate recovery of the underlying physical properties. Code and data are available at https://github.com/Zest86/ExPhy.

Accurate and robust time series forecasting is essential in many applications involving physical processes, such as manufacturing monitoring and astrophysical event detection. In these settings, predictive models must remain reliable under noise, variability, and measurement uncertainty while capturing temporally localized structures corresponding to physically meaningful events. Convolutional neural networks (CNNs) are widely used for such tasks due to their computational efficiency and strong representational capacity. However, their learned temporal representations often exhibit unstable or physically inconsistent attention patterns, reducing robustness, generalization, and interpretability. This paper introduces PhysAttNet, a physics-informed attention framework for time series forecasting. PhysAttNet augments a lightweight CNN forecaster with an attention head guided by domain-informed regularization reflecting the structural properties of physical signals. Specifically, three complementary constraints are imposed during training: an alignment regularization that encourages attention to follow smooth, peak-centered temporal structures derived from the input signal, a smoothness regularization that enforces continuous temporal evolution, and a sparsity regularization that promotes selective focus on informative intervals. These differentiable regularization terms introduce physics-guided inductive bias without requiring annotated explanations or manual supervision. Experiments on two distinct applications, namely predicting cutting forces during milling and forecasting flares in blazar time series, demonstrate that PhysAttNet improves forecasting accuracy, generalization, and prediction performance on structurally important events.

Multivariate forecasting in physical systems requires models that predict coupled temporal variables while preserving meaningful state evolution. Deep forecasters can fit temporal correlations, and physics-informed models can regularize predictions with scientific constraints, but these directions are often connected only at the decoded-output level. As a result, the hidden predictive state that generates future trajectories may remain statistically useful but physically unstructured. We introduce Phys-JEPA, a physics-informed joint-embedding predictive architecture for multivariate time-series forecasting. Phys-JEPA learns a latent world model in which predictive states are decomposed into physical and residual components, and physical consistency is imposed directly on latent states and latent transitions rather than only on decoded forecasts. This formulation uses known physical variables to organize the representation space while retaining residual capacity for unresolved dynamics. On Jena Climate 2009--2016, Phys-JEPA reduces aggregate MSE from 0.12482 to 0.12273 and temperature MSE from 0.01892 to 0.01831 at H=24. On Traffic, full Phys-JEPA improves aggregate MSE over the supervised baseline across all tested horizons, reducing H=192 MSE from 0.800784 to 0.773873. On Electricity, the best variant depends on horizon: static latent consistency is strongest at H=24 and H=48, while full Phys-JEPA gives the best aggregate and target-variable MSE at H=192. These initial results suggest that moving physics-informed learning from output space to latent predictive state space is a promising direction for interpretable temporal world models.

Most nowcasting systems, built on radar reflectivity, focus on current precipitation, ignoring the atmospheric precursors -- such as low-level convergence, turbulent eddies, and latent heating -- that offer a fleeting window to foresee storm birth. We introduce MeteoLogist, a physics-inspired radar intelligence framework that models the full life cycle of convection -- from its precursors to organized storm evolution. However, exploiting these precursors is non-trivial: they originate from multiple meteorological drivers -- thermodynamic, kinematic, and microphysical -- that evolve asynchronously (C1) and remain spatially fragmented (C2). To this end, MeteoLogist designs three tightly integrated components. The Physics-Tailored Encoders process radar echoes according to their intrinsic physical scales and semantics, forming thermodynamic, kinematic, and microphysical streams that capture distinct dynamical regimes. The Temporal-Phase Aligner addresses C1 by leveraging causal temporal attention to capture when and how different drivers interact and activate. The Cross-Field Spatial Aggregator addresses C2 through cross-regional fusion, aligning weak and scattered precursors across neighboring cells to expose upstream triggers and enforce spatial coherence. Evaluated on 3D-NEXRAD (2020--2022, US-wide), MeteoLogist boosts high-impact detection (CSI40) by +9.7% over strong baselines, and achieves a remarkable 37.67% gain during the storm-developing stage -- demonstrating true foresight in sensing storms before they appear. The code can be found in the supplementary material.

Time series forecasting under non-stationarity faces a fundamental tension between capturing stable representations and adapting to distribution shifts. Existing methods implicitly rely on static historical assumptions, leading to a critical failure mode we term Phase Amnesia, where models become blind to the evolving global context. To resolve this, we formalize non-stationary dynamics through three physical hypotheses: wold decomposition, dynamical phase evolution, and heteroscedastic manifold generation. These principles inspire PULSE, a physics-informed, plug-and-play framework adopting a Disentangle--Evolve--Simulate design philosophy. Specifically, PULSE utilizes phase-anchored disentanglement to resolve optimization interference caused by dominant trends, employs a Phase Router to actively generate future trajectories, and introduces Statistic-Aware Mixup (SAM) to ensure robustness against out-of-distribution volatility. Empirically, PULSE enables a simple MLP backbone to achieve state-of-the-art or highly competitive performance across 12 real-world benchmarks. This validates that a correct physics-informed inductive bias is far more critical than raw architectural complexity for non-stationary forecasting. The code is available at: https://github.com/Gemost/PULSE.

Autoregressive scientific forecasters often enforce physical or structural constraints by repairing each predicted state before feeding it back into the model. However, it remains unclear when stronger physical rule enforcement becomes reliable and when it becomes a source of distribution shift. We study this question through operator exactness, meaning whether the repair map is the identity on the target manifold and is aligned with the target geometry. We compare raw forecasting, post hoc repair, and in-loop repair across periodic incompressible Navier--Stokes, non-periodic CFDBench flows, and a hierarchical-forecasting support task. In the exact periodic regime, Fourier projection substantially improves rollout accuracy. On the NS-128 benchmark, a strong Raw-FNO has a final-step rollout MSE at horizon 100 of $(9.390 \pm 6.290)\times 10^{-5}$, and post hoc and in-loop projection reduce it to $(1.130 \pm 0.165)\times 10^{-6}$ and $(5.370 \pm 0.113)\times 10^{-7}$. However, once an exact projection is unavailable and only approximate boundary-preserving cleanup is available, the ordering changes. Across cavity, tube, dam, and cylinder flow, stronger Poisson-based cleanup can reduce divergence while worsening rollout error; target-distortion MSE predicts this harm far better than a linear-system residual. Controlled mismatch, screened cleanup, adaptive gating, and external-backbone checks show that the best approximate-regime operating point can be raw or near-identity. Hierarchical forecasting gives the same broader pattern. Exact forecast reconciliation is a stable baseline, whereas blended top-down repair, a validation-tuned interpolation toward historical-proportion top-down reconciliation, is dataset-dependent. Thus, constraint enforcement should be benchmarked by operator--data alignment before enforcement strength.

Operational weather prediction has long relied on physics-based numerical weather prediction (NWP), whose accuracy comes at the cost of substantial compute and complex simulation workflows. Recent transformer-based forecasters offer efficient data-driven alternatives, however transformers are physics-agnostic models. Additionally, standard transformer encoders evolve representations through discrete layer updates that may be less suited to modeling smooth latent dynamics. In this work, we propose a continuous-depth transformer encoder for weather forecasting that integrates Neural Ordinary Differential Equation (Neural ODE) dynamics within each encoder block. Specifically, we replace discrete residual updates with ODE-based updates solved using adaptive numerical integration. We also introduce a two-branch attention module that combines conventional patch-wise self-attention with an auxiliary branch that applies a derivative operator to attention logits, providing an additional change-sensitive interaction signal. To further align forecasts with governing principles, we propose a customized physics-informed training objective that enforces physical consistency as a soft constraint. We evaluate the proposed method against a standard discrete transformer baseline and an existing continuous-time Neural ODE forecasting variant, demonstrating the importance of PINN-Cast in short term weather forecasting.

AI data centers experience rapid fluctuations in power demand due to the heterogeneity of computational tasks that they have to support. For example, the power profile of inference and training of large language models (LLMs) is quite distinct and big divergences can result in the instability of the underlying electricity grid. In this paper we propose, to the best of our knowledge, the first physics-informed DLinear time-series model that can accurately forecast power utilization of an AI data center 5-80 minutes (short-term forecasting) into the future. The physics, based on a multi-node lumped thermal resistance-capacitance (RC) network consistent with Newton's law of cooling, is captured using newly derived time-dependent ordinary differential equations (ODE) that separately models and interlinks power consumption with the GPU compute and memory utilization and temperature. The resulting model, that we refer to as PI-DLinear, trained and evaluated on a real AI data center dataset and is not only more accurate than the state-of-the-art (SOTA) models tested, but the forecast profile respects the underlying physics under power throttling and load transient events. Relative to the SOTA transformer-based and non-transformer-based models, improvements in forecasting accuracy (averaged across all look-back and prediction windows) range from 0.782%-39.08% for MSE, 0.993%-51.82% for MAE, and 0.370%-22.28% for RMSE.

Accurate forecasting of industrial time series requires balancing predictive accuracy with physical plausibility under non-stationary operating conditions. Existing data-driven models often achieve strong statistical performance but struggle to respect regime-dependent interaction structures and transport delays inherent in real-world systems. To address this challenge, we propose DSPR (Dual-Stream Physics-Residual Networks), a forecasting framework that explicitly decouples stable temporal patterns from regime-dependent residual dynamics. The first stream models the statistical temporal evolution of individual variables. The second stream focuses on residual dynamics through two key mechanisms: an Adaptive Window module that estimates flow-dependent transport delays, and a Physics-Guided Dynamic Graph that incorporates physical priors to learn time-varying interaction structures while suppressing spurious correlations. Experiments on four industrial benchmarks spanning heterogeneous regimes demonstrate that DSPR consistently improves forecasting accuracy and robustness under regime shifts while maintaining strong physical plausibility. It achieves state-of-the-art predictive performance, with Mean Conservation Accuracy exceeding 99% and Total Variation Ratio reaching up to 97.2%. Beyond forecasting, the learned interaction structures and adaptive lags provide interpretable insights that are consistent with known domain mechanisms, such as flow-dependent transport delays and wind-to-power scaling behaviors. These results suggest that architectural decoupling with physics-consistent inductive biases offers an effective path toward trustworthy industrial time-series forecasting. Furthermore, DSPR's demonstrated robust performance in long-term industrial deployment bridges the gap between advanced forecasting models and trustworthy autonomous control systems.

Vision Transformers have achieved remarkable success in spatio-temporal prediction, but their scalability remains limited for ultra-high-resolution, continent-scale domains required in real-world environmental monitoring. A single European air-quality map at 1 km resolution comprises 29 million pixels, far beyond the limits of naive self-attention. We introduce CRAN-PM, a dual-branch Vision Transformer that leverages cross-resolution attention to efficiently fuse global meteorological data (25 km) with local high-resolution PM2.5 at the current time (1 km). Instead of including physically driven factors like temperature and topography as input, we further introduce elevation-aware self-attention and wind-guided cross-attention to force the network to learn physically consistent feature representations for PM2.5 forecasting. CRAN-PM is fully trainable and memory-efficient, generating the complete 29-million-pixel European map in 1.8 seconds on a single GPU. Evaluated on daily PM2.5 forecasting throughout Europe in 2022 (362 days, 2,971 European Environment Agency (EEA) stations), it reduces RMSE by 4.7% at T+1 and 10.7% at T+3 compared to the best single-scale baseline, while reducing bias in complex terrain by 36%.

Precipitation nowcasting (short-term forecasting) is still often performed using numerical solvers for physical equations, which are computationally expensive and make limited use of the large volumes of available weather data. Deep learning models have shown strong potential for precipitation nowcasting, offering both accuracy and computational efficiency. Among these models, convolutional neural networks (CNNs) are particularly effective for image-to-image prediction tasks. The SmaAt-UNet is a lightweight CNN based architecture that has demonstrated strong performance for precipitation nowcasting. This paper introduces the Multimodal Advection-Guided Small Attention GNet (MAD-SmaAt-GNet), which extends the core SmaAt-UNet by (i) incorporating an additional encoder to learn from multiple weather variables and (ii) integrating a physics-based advection component to ensure physically consistent predictions. We show that each extension individually improves rainfall forecasts and that their combination yields further gains. MAD-SmaAt-GNet reduces the mean squared error (MSE) by 8.9% compared with the baseline SmaAt-UNet for four-step precipitation forecasting up to four hours ahead. Additionally, experiments indicate that multimodal inputs are particularly beneficial for short lead times, while the advection-based component enhances performance across both short and long forecasting horizons.

Real physical systems are dissipative -- a pendulum slows, a circuit loses charge to heat -- and forecasting their dynamics from partial observations is a central challenge in scientific machine learning. We address the position-only (q-only) problem: given only generalized positions~$q_t$ at discrete times (momenta~$p_t$ latent), learn a structured model that (a)produces stable long-horizon forecasts and (b)recovers physically meaningful parameters when sufficient structure is provided. The port-Hamiltonian framework makes the conservative-dissipative split explicit via $\dot{x}=(J-R)\nabla H(x)$, guaranteeing $dH/dt\le 0$ when $R\succeq 0$. We introduce PHAST (Port-Hamiltonian Architecture for Structured Temporal dynamics), which decomposes the Hamiltonian into potential$V(q)$, mass$M(q)$, and damping~$D(q)$ across three knowledge regimes (KNOWN, PARTIAL, UNKNOWN), uses efficient low-rank PSD/SPD parameterizations, and advances dynamics with Strang splitting. Across thirteen q-only benchmarks spanning mechanical, electrical, molecular, thermal, gravitational, and ecological systems, PHAST achieves the best long-horizon forecasting among competitive baselines and enables physically meaningful parameter recovery when the regime provides sufficient anchors. We show that identification is fundamentally ill-posed without such anchors (gauge freedom), motivating a two-axis evaluation that separates forecasting stability from identifiability.