Process-Informed Forecasting Explained

• Process-Informed Forecasting is defined as forecasting that uses explicit process knowledge to structure predictions rather than relying solely on historical data.
• It integrates process structures, mechanistic priors, and process-specific covariates across areas like process mining, manufacturing, epidemics, and physical systems.
• The approach improves forecast plausibility and resilience under sparse data and nonstationarity while enforcing physically and operationally consistent predictions.

Process-Informed Forecasting (PIF) is a family of forecasting approaches in which predictive models are informed by explicit knowledge of the underlying process rather than relying only on historical observations. Across recent work, that process knowledge is injected through process-structured forecast targets, process-derived covariates, mechanistic or recipe-based priors, governing differential equations, probabilistic event-arrival models, and target-specific causal structure (Yu et al., 8 Dec 2025, Leribaux et al., 13 Oct 2025, Rubini et al., 24 Sep 2025, Miguelez et al., 2024). Taken together, these works suggest that PIF is best understood not as a single algorithmic family but as a design principle: forecasting is constrained, organized, or regularized by how the system actually operates.

1. Conceptual scope and recurring forms

The recent literature uses closely related but not identical terminology. In process mining, "Process Model Forecasting" (PMF) forecasts the future control-flow structure of a process by predicting the temporal evolution of directly-follows relations and reconstructing future directly-follows graphs (Yu et al., 8 Dec 2025). In pharmaceutical manufacturing, PIF is defined as a forecasting framework that augments standard time-series prediction with an explicit process-derived trajectory prior (Rubini et al., 24 Sep 2025). In epidemic forecasting, a multi-phase physics-informed neural network constrains neural trajectories with compartmental epidemic equations while allowing the mechanism to change across phases (Nguyen et al., 2024). In industrial process forecasting, a dual-level method first forecasts uncertain inputs probabilistically and then propagates them through a PINN constrained by process equations (Nasiri et al., 12 Jan 2026).

A plausible taxonomy suggested by these papers is that PIF appears in at least three recurrent forms.

Mode	Process knowledge source	Representative papers
Structured targets or covariates	event logs, actor behavior, operational context, dimensional feature maps	(Yu et al., 8 Dec 2025, Leribaux et al., 13 Oct 2025, Miguelez et al., 2024, Lampani et al., 23 Apr 2025)
Mechanistic priors or governing equations	recipe trajectories, ODE/PDE residuals, stochastic physical dynamics	(Rubini et al., 24 Sep 2025, Ma et al., 2020, Tipireddy et al., 2018, Nguyen et al., 2024, Nasiri et al., 12 Jan 2026)
Process-level stochastic or structural models	renewal processes, dynamics-informed generative processes, causal structure	(Turkmen et al., 2020, Cachay et al., 2023, Chen et al., 2024, Zhang et al., 22 May 2025)

The literature also rejects a narrow misconception that process-informed forecasting must always mean enforcing a full mechanistic simulator. Actor-enriched throughput forecasting injects process execution knowledge through actor-centric signals derived from event logs rather than through explicit process-model constraints (Leribaux et al., 13 Oct 2025). Physics-informed features inject domain knowledge at the representation layer through nonlinear, dimensionally homogeneous feature maps $\phi:\Omega \subset \mathbb{R}^m \to \mathcal{F}$, after which a standard supervised learner is fit in the transformed space (Lampani et al., 23 Apr 2025). Conversely, the strongest-form PIF papers derive the probabilistic prior itself from governing stochastic equations, as in physics-informed Gaussian process regression for power grids (Ma et al., 2020).

2. Process-structured targets and representations

A central branch of PIF uses process structure not merely as side information but as the forecast target itself. In PMF, the starting point is an event log

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$

where each case

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$

is an ordered event sequence. Predictive Process Monitoring is contrasted with PMF by the mapping

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$

from a case prefix to a target such as next activity or remaining time, whereas PMF targets system-level dynamics, namely the evolution of the process model itself (Yu et al., 8 Dec 2025).

The process-aware representation is built from directly-follows relations. For activity labels $a_i$ and $a_j$, the directly-follows count is

$>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$

and the corresponding directly-follows graph is

$DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$

After partitioning the event log into equal-length windows $\Delta T$, one obtains time-indexed DFG snapshots and then one time series per edge,

$\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$

Future process models are forecast either at graph level,

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$0

or equivalently at DF-series level and then reassembled into forecasted DFGs (Yu et al., 8 Dec 2025). This is a direct instance of process-informed forecasting because the target variables are process relations and the predicted object remains an interpretable process model.

A second representation-level formulation appears in actor-enriched forecasting of process performance. There the target is the daily average throughput time of cases that start on day $$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$1,

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$2

with case throughput time

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$3

What makes the model process-informed is the addition of actor-centric temporal signals extracted from same-case transitions. Consecutive event pairs are assigned one of four actor behavior types,

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$4

namely continuation, interruption, handover to idle, and handover to busy. For each day $$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$5 and behavior $$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$6, the paper defines daily counts

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$7

and daily total durations

$$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$8

These series are then combined with lagged and rolling features to forecast process-level throughput performance (Leribaux et al., 13 Oct 2025). Here the process-informed element is neither a simulator nor an explicit process model; it is a feature-level encoding of execution semantics internal to the process.

Industrial operations forecasting provides a third representation. An adaptive IO-HMM forecasts process time variables such as Operating Time $$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$9, Net Operating Time $$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$0, and Valuable Time $$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$1, which are themselves embedded in production-time accounting relations,

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$2

The model conditions state transitions on process context such as shift type and continuous responses on shift, ideal cycle speed, indicators for the first observation of a shift or production order, and lagged responses (Miguelez et al., 2024). This suggests that PIF can be realized by choosing targets and covariates that already encode process semantics before any forecasting model is applied.

A further representation-layer variant is physics-informed feature construction. There, nonlinear feature maps are designed so that transformed features are physically meaningful and dimensionally homogeneous, often with the same physical dimension as the label. The paper’s illustrative example,

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$3

shows how domain knowledge can enter before model fitting, after which a regularized linear model is fit in transformed space,

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$4

with ridge solution

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$5

This is not a temporal dynamics model by itself, but it establishes an important PIF principle: process knowledge can be injected through the representation layer as well as through dynamics (Lampani et al., 23 Apr 2025).

3. Mechanistic priors, probabilistic process models, and physical consistency

A stronger form of PIF uses explicit process laws or process-derived priors. In pharmaceutical lyophilization, the process prior is an idealized piecewise-linear temperature trajectory derived from the manufacturing recipe, with stage setpoints $$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$6 and phase-transition times $$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$7. This prior is not a full first-principles thermodynamic model; it is the intended thermal trajectory of freezing, primary drying, and secondary drying (Rubini et al., 24 Sep 2025). For classical forecasters, the process-informed forecast is a convex combination

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$8

while for neural models the prior enters through additional loss terms. The paper studies three formulations: fixed weighting,

$$\sigma_i = \langle e_{i,1}, e_{i,2}, \dots, e_{i,T_i}\rangle$$9

uncertainty-based weighting,

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$0

and Residual-Based Attention. The paper evaluates not only RMSE but also gradient-based plausibility metrics, thereby making physical consistency a first-class forecasting objective (Rubini et al., 24 Sep 2025).

Physics-informed Gaussian process regression for power grids is an even stronger formulation because the prior mean and covariance are computed from stochastic equations governing power-grid dynamics rather than fit from data by generic kernel learning. In the three-generator case, the dynamics are governed by the swing equations

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$1

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$2

with stochastic wind mechanical power modeled as an Ornstein–Uhlenbeck process. Monte Carlo simulation of these stochastic equations yields empirical means and cross-covariances, which define the multivariate GP prior. Posterior forecasting and latent-state estimation then follow from standard Gaussian conditioning. This construction allows forecasting not only observed angles and angular speeds but also unobserved wind mechanical power (Ma et al., 2020). The earlier single-generator version uses the same principle in a simpler swing-equation setting and shows that the forecast horizon depends on the input correlation time and the system relaxation time (Tipireddy et al., 2018).

Intermittent demand forecasting with renewal processes provides a different process-model view. Instead of treating intermittent demand as a sparse scalar series, the paper decomposes it into interdemand intervals $f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$3 and positive sizes $f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$4, where

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$5

Demand occurrence is modeled as a discrete-time renewal process, and one-step forecast structure is expressed as

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$6

that is, forecasted demand equals the arrival hazard at current age times expected demand size (Turkmen et al., 2020). This is process-informed because it explicitly models when events occur and how large they are once they occur. It also generalizes Croston-type methods and admits neural parameterizations by replacing exponential smoothing recursions with LSTMs.

These works share a common premise: process knowledge is most consequential when forecasting must remain physically or operationally plausible under sparse data, nonstationarity, or partial observability. They also show that process-informedness can mean different things—trajectory priors, governing equations, or event-arrival laws—provided the forecasting distribution is shaped by process structure rather than solely by empirical correlation.

4. Hybrid and adaptive architectures

A prominent recent trend is to combine flexible neural approximators with process constraints in architectures that are explicitly adaptive. MP-PINN forecasts epidemic waves by fitting neural functions for $f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$7 and $f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$8 while penalizing violations of the SIR equations

$f_\theta : \Sigma^{*} \rightarrow \mathcal{Y},$9

Single-phase PINN uses one parameter pair $a_i$0, whereas multi-phase PINN uses phase-specific $a_i$1 and phase start times $a_i$2. In the reported experiments, training uses the first $a_i$3 days of each COVID-19 wave and forecasting covers the remaining $a_i$4 days; for $a_i$5, the second phase begins at $a_i$6 (Nguyen et al., 2024). This architecture preserves a mechanistic epidemic prior while allowing process parameters to change over time in response to interventions or behavioral shifts.

The dual-level physics-informed framework for multi-step forecasting makes a different architectural split. Level 1 forecasts future input variables with hybrid LSTM-state-transition models, while Level 2 feeds those stochastically forecast inputs into a PINN that enforces output dynamics. The generic recursive structure is

$a_i$7

with recursive use of prior forecasts over the horizon, and the output PINN is built from discrete-time Runge–Kutta residuals. The case studies include a continuous stirred tank reactor, an axial-dispersion plug-flow reactor, and an industrial froth flotation process (Nasiri et al., 12 Jan 2026). This architecture is notable because it separates two distinct forecasting difficulties: uncertainty in future exogenous drivers and adherence of outputs to known process equations.

Two adjacent literatures extend the boundary of PIF without fully occupying its mechanistic core. DYffusion replaces noise-based diffusion trajectories with temporally meaningful interpolants between current and future states. Its learned process evolves through

$a_i$8

so the generative path follows the structure of temporal dynamics rather than generic Gaussian corruption (Cachay et al., 2023). This is described as dynamics-informed rather than fully physics-informed, but it is relevant because it injects temporal process structure into the generative mechanism itself.

Probabilistic forecasting with stochastic interpolants and Föllmer processes is similar in spirit. It constructs an artificial SDE

$a_i$9

such that $a_j$0 has the correct conditional future-state distribution given current state $a_j$1. The paper proves that the drift is non-singular even though the initial condition is a point mass, and derives a Föllmer-process choice of diffusion that minimizes sensitivity to drift-estimation error in path-space KL divergence (Chen et al., 2024). The process is explicitly fictitious rather than physical, but it remains a process-level forecast dynamics in state space.

CAIFormer occupies an even more cautious boundary position. It constructs a structural causal model from observational data, partitions each target’s historical variables into endogenous, direct causal, collider causal, and spurious-correlation sub-segments, and excludes the spurious segment from prediction (Zhang et al., 22 May 2025). The paper itself is best interpreted as causal-informed or structure-informed forecasting rather than classical mechanistic PIF, but it shows that process structure can also be operationalized as target-specific dependency structure.

5. Empirical evidence and evaluation practices

Empirical results across these papers support the general usefulness of process-informed structure, but they also show that the gains depend on what is injected and how it is evaluated. In PMF with time-series foundation models, zero-shot TSFMs already outperform strong PMF baselines on most datasets and metrics. For MAE, the strongest zero-shot values are $a_j$2 on BPI2017, $a_j$3 on BPI2019_1, $a_j$4 on Sepsis, and roughly $a_j$5–$a_j$6 on Hospital Billing, corresponding to reductions of $a_j$7, $a_j$8, $a_j$9, and about $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$0–$>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$1 against the best baselines. For RMSE, the best zero-shot values are $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$2, $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$3, $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$4, and about $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$5–$>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$6, giving reductions of $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$7, $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$8, $>_{\mathcal{L}(a_i,a_j) \in \mathbb{N},$9, and about $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$0 (Yu et al., 8 Dec 2025). Fine-tuning can help, but improvements are often marginal, dataset-dependent, or unstable.

That same PMF study also demonstrates why process-aware evaluation matters. In addition to MAE and RMSE, it uses Entropic Relevance (ER), where lower ER means a forecasted DFG encodes observed log behavior more concisely and accurately. ER is generally comparable to classical baselines on BPI2017, BPI2019_1, and Hospital Billing, but much worse on Sepsis, where pointwise DF forecasts remain numerically strong while the reconstructed process models have poor fitting ratios (Yu et al., 8 Dec 2025). This establishes a core PIF principle: better scalar forecast errors do not automatically imply behaviorally plausible process models.

Actor-enriched forecasting likewise reports consistent gains from process-derived signals. Relative to throughput-time-only baselines, actor-enriched models improve RMSE, MAE, and $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$1 across BPIC2017 and BPIC2012, with especially strong improvements in BPIC2012. For example, XGBoost RMSE improves from $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$2 to $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$3 in BPIC2012, and several deep models also show multi-hour absolute gains (Leribaux et al., 13 Oct 2025). The feature-importance results are also informative: autoregressive throughput features remain dominant, but actor-derived variables such as $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$4, $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$5, and rolling interruption-duration features rank highly, showing that process-derived covariates explain variation not recoverable from the target history alone.

In pharmaceutical lyophilization, the evaluation explicitly separates accuracy, plausibility, and robustness. Classical PIF variants tend to improve gradient-based plausibility while sometimes worsening RMSE. For neural models, especially at larger capacities, PIF often improves both accuracy and plausibility. At around $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$6k parameters, examples include $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$7 versus $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$8 RMSE for RNN_uncertainty versus standard RNN and $DFG_{\mathcal{L}} = (V,E), \quad V = \{a_i \in \mathcal{A}\}, \quad E = \{(a_i,a_j,w_{ij}) \mid w_{ij} = >_{\mathcal{L}(a_i,a_j)\}.$9 or $\Delta T$0 versus $\Delta T$1 for process-informed Transformer variants versus standard Transformer (Rubini et al., 24 Sep 2025). The robustness experiments further show that process-informed loss shaping helps, but architecture remains a primary determinant of noise resilience.

MP-PINN provides a particularly clear comparison among mechanistic, data-driven, and hybrid process-informed models. Using sMAPE over $\Delta T$2 Italian regions, the next-30-day scores are $\Delta T$3 for SIR, $\Delta T$4 for MLP, $\Delta T$5 for SP-PINN, and $\Delta T$6 for MP-PINN. Beyond 30 days, the scores are $\Delta T$7, $\Delta T$8, $\Delta T$9, and $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$0, respectively; over all forecast days they are $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$1, $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$2, $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$3, and $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$4 (Nguyen et al., 2024). The pattern is interpretable: single-phase process-informed learning is strongest short-term, while multi-phase process-informed learning is strongest long-term.

The dual-level industrial framework shows similar evidence in physical process settings. On output forecasting with known inputs, PINNs achieve substantially lower test MSE than purely data-driven FFNNs: $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$5 versus $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$6 for CSTR, $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$7 versus $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$8 for ADPFR, and $\{>_{\mathcal{L}_{t_i}(a_p,a_q)\}_{i=1}^{T}.$9 versus $$\mathcal{L} = \{\sigma_1, \sigma_2, \dots, \sigma_N\},$$00 for flotation (Nasiri et al., 12 Jan 2026). In the integrated two-level pipeline, the best combinations are PINN with hybrid input forecasting models, especially hybrid Matérn or hybrid exponential depending on the case. The reported outputs include MSE, log-likelihood, and confidence intervals, so both point accuracy and probabilistic quality are evaluated.

6. Limitations, boundary cases, and open directions

The literature is equally clear that PIF is not universally superior and that its failures are often diagnostic. A comparison study on the Lorenz-3 system shows that when the filter is provided with complete and accurate information, the process-based forecaster dominates the data-based method, but when parameters must be learned and noise is misspecified, the SVM forecaster outperforms the filter (Bornn et al., 2012). This is a fundamental caution for PIF: process knowledge is a powerful inductive bias only to the extent that it is accurate enough to reduce variance more than it introduces bias.

Several domain-specific limitations recur. PMF with TSFMs evaluates only four event logs, studies a limited set of PEFT methods, and leaves open whether better multivariate or graph-aware TSFM adaptations could exploit inter-DF dependencies more effectively (Yu et al., 8 Dec 2025). Actor-enriched throughput forecasting aggregates at daily resolution, learns on a smoothed differenced target, and depends on reliable resource identifiers and timestamped event sequences (Leribaux et al., 13 Oct 2025). Lyophilization PIF depends strongly on architecture and weighting strategy; no single loss formulation is universally best, and robustness rankings change across datasets (Rubini et al., 24 Sep 2025).

Boundary cases also matter for conceptual clarity. CAIFormer is explicitly better described as causal-informed or structure-informed forecasting than as a classical mechanistic PIF method, because its process knowledge comes from an estimated causal DAG rather than from governing equations (Zhang et al., 22 May 2025). DYffusion is dynamics-informed rather than physics-informed because it redesigns the generative process around temporal evolution without imposing explicit physical laws (Cachay et al., 2023). Physics-informed features improve supervised learning and interpretability, but the paper itself notes that the framework is mostly static, does not learn explicit temporal dynamics, and provides limited causal guarantees (Lampani et al., 23 Apr 2025). These edge cases show that PIF has porous boundaries; the concept shades into causal forecasting, dynamics-informed generative modeling, and feature-level scientific machine learning.

Open directions named in the cited works are notably consistent. PMF points to richer structural representations beyond directly-follows relations, more TSFMs, more PEFT methods, more high-quality logs, and integration of forecasts into interactive process mining tools (Yu et al., 8 Dec 2025). MP-PINN identifies automatic phase detection as a next step (Nguyen et al., 2024). The stochastic-interpolant forecasting framework explicitly proposes incorporation of physical structure into the generative model as future work (Chen et al., 2024). A plausible implication is that future PIF systems will become more modular: probabilistic input forecasting, process-aware state representation, mechanistic or structural priors, and process-aware evaluation will increasingly be combined rather than treated as mutually exclusive alternatives.

In its broadest contemporary sense, Process-Informed Forecasting therefore denotes forecasting that remains accountable to how a system is organized and evolves. In process mining, it forecasts future process structure; in industrial operations, it forecasts time-loss variables and process KPIs using operational context; in physical and biochemical systems, it combines learned predictors with recipe trajectories, balance laws, ODEs, PDEs, or stochastic governing equations. The unifying feature is not any single model class, but the decision to make forecast distributions, forecast targets, or forecast features reflect the process itself.