PCFM

PCFM — Probabilistic Causal Foundation Model

PCFM is a topology-constrained, any-variate time-series foundation model for industrial process telemetry. Unlike existing time-series foundation models, it accepts a directed, lag-annotated causal graph G(A, T) — the physical topology of a plant — as an inference-time structural prompt: a GNN-style spatial router restricts cross-variate information flow to the graph's edges and reads each driver at its physical cause–effect lag, at a spatial cost linear in the edge count. Its learned parameters are independent of the number of input variates, so one set of pre-trained weights serves arbitrary sensor configurations.

Pre-trained with a unified masked-reconstruction objective on a purpose-built, topology-annotated corpus (103 billion timepoints across 12.8 million lag-annotated sparse directed graphs), the identical weights execute — zero-shot, selected purely through the observation mask — univariate, multivariate, and covariate-informed forecasting, temporal imputation, and structural what-if (interventional) simulation, emitting probabilistic bounds through a nine-level quantile head.

Results

Note: the results reported here are without test-time ensembling and match the dissertation.

• GIFT-Eval (97 tasks, zero-shot): relative CRPS 0.493 overall (geometric mean vs. Seasonal Naive) — fifth among the models compared, behind only TiRex-2, Chronos-2, TiRex, and TimesFM-2.5. On the multivariate slice the default sparse-graph (CP) prompt attains the best score of the comparison (0.448), ahead of Chronos-2, TiRex-2, and TimesFM-2.5, and the best relative MAE (0.671) as well.
• fev-bench (100 tasks, 46 with covariates, zero-shot): third-highest skill score of the comparison under both headline metrics — MASE (30.3%) and SQL (37.6%) — behind only Chronos-2 and TiRex-2.

Pairwise skill scores on fev-bench under MASE (row model vs. column model as baseline; bootstrapped 95% CIs in parentheses).

Files

• model.safetensors — the pre-trained checkpoint (continued pre-training of the published paper's checkpoint on the identical corpus, same five-stage curriculum, peak learning rate reduced to one tenth). Loading requires the model code from the PCFM GitHub repository.
• Gifteval_CRPS_Results.txt — GIFT-Eval leaderboard (relative CRPS; full / univariate / multivariate and variate-type × horizon breakdowns).
• Fevbench_MASE_Results.csv, Fevbench_SQL_Results.csv — fev-bench leaderboards (win rates and skill scores).
• Fevbench_MASE_Skillscore.pdf, Fevbench_SQL_Skillscore.pdf — pairwise skill-score matrices.

Related artifacts

• Code (model, evaluation harness, what-if benchmark suite): https://github.com/software-competence-center-hagenberg/pcfm
• Pre-training corpus (Apache Arrow shards with state tensors, adjacency and lag matrices): Google Drive folder

Publications

Main reference (dissertation): Michael Mayr. Towards Foundational Digital Twins in Process Industry: Topology-Constrained Time-Series Foundation Models. PhD thesis, Johannes Kepler University Linz, 2026. https://resolver.obvsg.at/urn:nbn:at:at-ubl:1-108719

The checkpoint in this repository is the one reported in the dissertation: the model first described in the DaWaK paper below, with continued pre-training.

Conference paper: Michael Mayr and Georgios C. Chasparis. Topologically-Constrained Any-Variate Time-Series Foundation Models for Twinning of Continuous Industrial Processes. In: Big Data Analytics and Knowledge Discovery (DaWaK 2026), LNCS 16861, Springer, pp. 189–196. DOI: 10.1007/978-3-032-34896-8_15

@phdthesis{Mayr2026Thesis,
  author    = {Mayr, Michael},
  title     = {Towards Foundational Digital Twins in Process Industry: Topology-Constrained Time-Series Foundation Models},
  school    = {Johannes Kepler University Linz},
  year      = {2026},
  url       = {https://resolver.obvsg.at/urn:nbn:at:at-ubl:1-108719}
}

@inproceedings{MayrChasparis2026,
  author    = {Mayr, Michael and Chasparis, Georgios C.},
  title     = {Topologically-Constrained Any-Variate Time-Series Foundation Models for Twinning of Continuous Industrial Processes},
  booktitle = {Big Data Analytics and Knowledge Discovery (DaWaK 2026)},
  series    = {Lecture Notes in Computer Science},
  volume    = {16861},
  pages     = {189--196},
  publisher = {Springer},
  year      = {2026},
  doi       = {10.1007/978-3-032-34896-8_15}
}