A Pretrained Model for Bayesian Filtering in Heterogeneous Dynamical Systems

Ke Fan; Wen Wu; Haitao Su; Yilin Wang; Yuan Zhou; Chao Zhang

cs.LGOct 29, 2026

A Pretrained Model for Bayesian Filtering in Heterogeneous Dynamical Systems

Authors: Ke Fan, Wen Wu, Haitao Su, Yilin Wang, Yuan Zhou, Chao Zhang

Organizations: Shanghai AI Laboratory · Tsinghua University

Abstract

Filtering has not yet had its foundation-model moment: state estimation is still dominated either by online Bayesian filtering algorithms that require known dy- namics as inputs, or by learning-based neural filters trained from scratch for a single system or parametric family. The core challenge is structural heterogene- ity: dynamical systems differ in their governing dynamics, physical scales, and noise levels, precluding a shared representation space for a reusable filter. To bridge this gap, we introduce FilterFM, the first pretrained model for Bayesian filtering that estimates latent states across heterogeneous dynamical systems with a single checkpoint. FilterFM employs a hierarchical two-level observation se- quence encoder and a prior-to-posterior state-reconstruction module that together handle varying observation and state dimensions across heterogeneous systems, while modeling state uncertainty via quantile prediction. Trained on a mixture of heterogeneous filtering tasks, FilterFM exhibits superior in-domain performance, delivering significant gains in estimation accuracy, runtime efficiency, and noise robustness compared to existing baselines. Beyond in-domain performance, it marks the first model to successfully enable zero-shot filtering on unseen dynamics, bridging the generalization gap in state estimation. Furthermore, we uncover clear scaling behaviors in foundation-model-based Bayesian filtering, which combined with comprehensive component ablation studies, validate the structural optimality of our design and broaden the generalization scope for this research field.

Figures & tables

Figure 1: Generalization scopes of FilterFM, LLMFilter, and FM-UKF. LLMFilter trains on one system at a time and transfers across datasets at fixed state dimension

N

via System-as-Prompt. FM-UKF learns dynamics across parameter settings within one system family and uses a UKF with a known observation model. FilterFM is pretrained on a corpus of heterogeneous systems and evaluated on unseen systems using the same checkpoint without fine-tuning.

Figure 2: FilterFM architecture. The encoder combines patch-level temporal and variable interaction with intra-patch temporal interaction, cross-scale conditioning, and intra-patch variable interaction. State queries read shifted and current local memory for prior prediction and posterior update, followed by quantile decoding and de-normalization to the original state scale.

Method	Ours	Learning-Based Filters							Online Bayesian Filters
	FilterFM	LLMFilter	FM-UKF	MEstimator	RStateNet	ProTran	KalmanNet	RNNYYF	EnKF	PF	EnKFI	EnKFS	HubEnKF
Tracking	0.9683	1.9944	1.2556	2.7357	17.8542	1.5862	17.7382	18.7570	1.7846	1.7820	1.7858	1.7874	1.7853
Pendulum	0.0840	0.0891	0.1318	0.2234	0.2243	0.1095	0.1457	2.1915	0.0822	0.3266	0.2776	0.0933	0.0897
Selkov	0.0155	0.0189	0.0165	0.0264	0.0336	0.0182	0.1059	0.2291	0.0165	0.0167	0.1384	0.0217	0.0219
Oscillator	0.0166	0.0166	0.0171	0.0249	0.0320	0.0181	2.9264	0.1307	0.0170	0.0172	0.1313	0.0206	0.0212
Hopf	0.0156	0.0176	0.0167	0.0280	0.0304	0.0176	0.0540	0.1991	0.0164	0.0169	0.2095	0.0212	0.0216

Table 1: In-domain raw-space RMSE on 14 dynamical systems (lower is better). Red, blue underlining, and teal italics indicate the top three results;

1^{st}

Count denotes row-wise wins. Rankings use unrounded values. Missing entries (–) are excluded from the corresponding method’s average.

Figure 3: State-estimation trajectories on five high-dimensional in-domain systems (columns), with three state channels per system (rows). FilterFM tracks rapid changes in Lorenz96 and VL20, turning points in Vanderpol64, and small fluctuations in Rossler96. LLMFilter shows jumps and large offsets on Cubic-Sensoring100, while FM-UKF exhibits timing and amplitude errors. Light-red shading shows FilterFM’s predicted

0.1

–

0.9

quantile interval.

Figure 4: Per-step inference latency on 14 benchmark systems. The y-axis reports latency in ms/step on a base-10 logarithmic scale. Measurements use a single NVIDIA A800 GPU and exclude model loading. FilterFM is fastest on every system, averaging 0.145 ms/step.

Figure 5: Observation-noise misspecification: true covariance is scaled up while online filters retain their original assumed covariance. FilterFM achieves the lowest pooled RMSE in 38 of the 40 displayed settings across low- and high-dimensional systems. Curves show pooled RMSE; shading denotes the population standard deviation of trajectory-wise RMSE over 20 test trajectories.

Figure 6: Ablation study of five architectural variants. Left and middle show raw and normalized RMSE distributions on in-domain and out-of-domain systems, respectively; boxes mark interquartile ranges and medians, with 5th–95th percentile whiskers. Right shows training-loss curves under identical data, optimization settings, and 20-epoch budgets. The full model achieves the lowest median errors on both splits and the lowest final training loss.

Figure 7: Prior-loss sweep and scaling behavior. (A) Raw RMSE on nine high-dimensional in-domain systems (blue; 40–100 state variables) and OOS systems (red). Dashed lines: direct posterior reconstruction; star: best OOS setting (

\lambda_{\rm prior}=1.0

). Full in-domain results can be found in Appendix C.3.2 . (B,C) Mean and median OOS normalized RMSE versus training-system count and model capacity, with measurements (dots), power-law fits (dashed curves), and logarithmic x-axes.

Appendix figures & tables47 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Window length	Batch size	Learning rate	Maximum training epochs
MEstimator	40	16	$10^{-4}$	50
RStateNet	40	16	$10^{-4}$	50
ProTran	40	16	$10^{-4}$	50
KalmanNet	40	1	$10^{-4}$	50
RNNYYF	200	16	$4\times 10^{-4}$	50
LLMFilter	40	1	$10^{-4}$	10

Appendix

Table 2: Baseline training settings for Table 1 . Window lengths are measured in time steps,

N

denotes state dimension, and maximum training epochs is the preset upper limit for each run.

System	System type	State / Obs. Dim.	$\Delta t$	Integrator	$\bm{R}$	$\bm{Q}$
Tracking ( Smith et al., 1962 )	Linear target tracking	$4/4$	$0.1$	Forward Euler	$0.1\,\mathbf{I}_{4}$	$10\,\mathbf{I}_{4}$
Double-Pendulum ( Levien & Tan, 1993 )	Chaotic rigid-body mechanics	$4/4$	$0.01$	RK4	$10^{-4}\mathbf{I}_{4}$	$0.1\,\mathbf{I}_{4}$
Selkov ( Sel’kov, 1968 )	Biochemical glycolytic oscillation	$2/2$	$0.01$	RK4	$10^{-4}\mathbf{I}_{2}$	$10^{-3}\mathbf{I}_{2}$
Oscillator ( Brunton et al., 2016 )	Canonical nonlinear oscillation	$2/2$	$0.01$	RK4	$10^{-4}\mathbf{I}_{2}$	$10^{-3}\mathbf{I}_{2}$
Hopf ( Hassard et al., 1981 )	Normal-form bifurcation oscillator	$2/2$	$0.01$	RK4	$10^{-4}\mathbf{I}_{2}$	$10^{-3}\mathbf{I}_{2}$
Lorenz96 ( Lorenz, 1995 )	Chaotic atmospheric dynamics	$72/72$	$0.01$	RK4	$10^{-4}\mathbf{I}_{72}$	$10\,\mathbf{I}_{72}$

Appendix

Table 3: Configuration of the fourteen in-domain benchmark systems. Here

\bm{R}

and

\bm{Q}

denote the process and observation noise covariances used for testing, with 20 test trajectories per system.

Data source	Training	Held-out	Total
In-domain benchmark (14 systems)	6,720	280	7,000
Panda low-dimensional (790 systems)	3,360	140	3,500
Panda high-dimensional composites	5,040	200	5,240
Total for training domains	15,120	620	15,740
Panda zero-shot (457 systems)	0	8,207	8,207

Appendix

Table 4: Composition of the main training corpus and held-out evaluation sets. All trajectories contain 200 time steps.

Figure 8: State-dimension distributions for (A) the fourteen benchmark systems and (B) the low-dimensional and zero-shot subsets in Table 4 . (C) The 15,120 training trajectories, stacked by data source. All horizontal axes display state dimension

N

as discrete categories.

Dataset	FilterFM	LLMFilter	FM-UKF	EnKF	PF
Tracking	0.1063	1.4186	1.8507	0.5247	0.3518
Pendulum	0.1062	1.5954	1.8457	0.8383	0.6614
Selkov	0.1067	1.4057	1.8379	0.4834	0.3767
Oscillator	0.1089	1.8586	1.8305	0.7892	0.6796
Hopf	0.1122	2.2081	1.8338	0.5068	0.3949
Lorenz96	0.1175	6.5480	19.0996	9.9052	4.8176

Appendix

Table 5: Runtime details for the 14-system benchmark. All methods are evaluated on a single NVIDIA A800 GPU. Values are measured in ms/step with model loading excluded. LLMFilter timing includes the full System-as-Prompt input. Lower is better.

Dataset	Method	OCER=1	OCER=25	OCER=50	OCER=75	OCER=100
Tracking	FilterFM	$\mathbf{0.968{\scriptscriptstyle\pm 0.118}}$	$\mathbf{3.028{\scriptscriptstyle\pm 0.579}}$	$\mathbf{4.337{\scriptscriptstyle\pm 0.868}}$	$\mathbf{5.257{\scriptscriptstyle\pm 1.091}}$	$\mathbf{5.972{\scriptscriptstyle\pm 1.219}}$
	LLM-Filter	$1.994{\scriptscriptstyle\pm 0.672}$	$8.839{\scriptscriptstyle\pm 3.140}$	$11.438{\scriptscriptstyle\pm 4.237}$	$12.902{\scriptscriptstyle\pm 4.844}$	$13.921{\scriptscriptstyle\pm 5.307}$
	EnKF	$1.785{\scriptscriptstyle\pm 1.020}$	$4.300{\scriptscriptstyle\pm 0.582}$	$5.865{\scriptscriptstyle\pm 0.564}$	$7.094{\scriptscriptstyle\pm 0.608}$	$8.134{\scriptscriptstyle\pm 0.677}$
	EnKFS	$1.787{\scriptscriptstyle\pm 1.023}$	$3.938{\scriptscriptstyle\pm 0.679}$	$5.193{\scriptscriptstyle\pm 0.726}$	$6.154{\scriptscriptstyle\pm 0.828}$	$6.949{\scriptscriptstyle\pm 0.932}$
	HubEnKF	$1.785{\scriptscriptstyle\pm 1.018}$	$4.193{\scriptscriptstyle\pm 0.615}$	$5.598{\scriptscriptstyle\pm 0.664}$	$6.649{\scriptscriptstyle\pm 0.767}$	$7.502{\scriptscriptstyle\pm 0.877}$
Selkov	FilterFM	$\mathbf{0.0155{\scriptscriptstyle\pm 0.0010}}$	$\mathbf{0.0405{\scriptscriptstyle\pm 0.0046}}$	$\mathbf{0.0512{\scriptscriptstyle\pm 0.0063}}$	$\mathbf{0.0591{\scriptscriptstyle\pm 0.0074}}$	$\mathbf{0.0659{\scriptscriptstyle\pm 0.0083}}$

Appendix

Table 6: Model-mismatch raw-space RMSE on the 20 held-out testing trajectories, reported as pooled RMSE

{\scriptscriptstyle\pm}

the population standard deviation of trajectory-wise RMSE. FilterFM rows are shaded. Lower is better, and the lowest pooled RMSE for each dataset and OCER is bolded.

Figure 9: Ablation trajectories. In-domain and zero-shot system trajectories across five architectural variants.

Update rule	$\bm{\lambda_{\rm prior}}$	In-domain raw	High-dim raw	OOS raw
Direct posterior	–	0.2351	0.2428	1.1055
Prior-to-posterior	0.05	0.2317	0.2413	1.0982
Prior-to-posterior	0.20	0.2334	0.2419	1.1021
Prior-to-posterior	0.50	0.2313	0.2403	1.0953
Prior-to-posterior	1.00	0.2317	0.2390	1.0660
Prior-to-posterior	1.50	0.2321	0.2396	1.0786

Appendix

Table 7: Prior-loss sweep and direct-posterior ablation. In-domain raw averages over all fourteen benchmark systems; High-dim raw averages over the subset with state dimension

\geq 40

; OOS raw averages over zero-shot systems. Lower is better.

IC	$\bm{N_{\rm sys}}$	Train trajectories	OOS normalized median	OOS normalized mean
128	164	19,101	0.2624	0.3086
64	328	19,806	0.2417	0.2804
32	656	19,672	0.2270	0.2661
16	1312	19,534	0.2174	0.2554
8	2623	19,369	0.2102	0.2479
4	5245	19,357	0.2050	0.2426

Appendix

Table 8: System-diversity scaling details. IC is the target number of initial conditions per system, and

N_{\rm sys}

is the nominal system count of the corresponding training subset. The actual training-trajectory counts reflect data filtering and validation allocation. Values are OOS normalized RMSE statistics over 200-step windows sliced from 4096-step held-out trajectories. Lower is better.

Parameters	Training loss at epoch 10	In-domain validation raw RMSE	OOS raw RMSE
			Mean	Median
2.89M	0.148945	0.284954	0.860030	0.218894
5.09M	0.143718	0.277112	0.847594	0.216160
11.36M	0.135006	0.266257	0.833914	0.213774
20.11M	0.129503	0.263754	0.829695	0.212346
34.54M	0.125071	0.257718	0.817275	0.210798

Appendix

Table 9: Model-parameter scaling training and raw-space evaluation results. All results use the same data, training budget, and epoch-10 checkpoint selection and are from single-seed runs. Lower is better; the best value in each metric column is bolded.

Parameters	Hidden width $\bm{d}$	FFN hidden width	Self-attention heads	Patch-summary temporal encoding layers $\bm{J}_{\rm enc}$	Patch-level blocks	Intra-patch blocks	Prior-to-posterior state-reconstruction blocks
2,888,007	96	384	4	2	3	2	1
5,091,751	128	512	4	2	3	2	1
11,360,871	192	768	4	2	3	2	1
20,112,167	256	1024	4	2	3	2	1
34,541,847	336	1344	4	2	3	2	1
51,205,431	336	1344	6	3	4	3	2

Appendix

Table 10: Architecture configurations for model-capacity scaling. Patch-summary temporal encoding layers process observation tokens before pooling, as described in Appendix D.2 . Patch-level blocks combine causal temporal interaction with prototype-based variable interaction; intra-patch blocks additionally apply cross-scale conditioning between these operations, as detailed in Appendices D.3 – D.4 . Prior-to-posterior state-reconstruction blocks follow Section 4.2 and Appendix D.5 . The listed head count applies to temporal self-attention for patch-summary construction, patch-level and intra-patch temporal self-attention, and self-attention among prototypes at both levels. Gated reassembly and the Retrieve operators use single-head cross-attention.

Figure 10: Training-trajectory scaling at fixed model capacity and system diversity. Points show OOS trajectory-level mean or median RMSE averaged over three seeds, with vertical error bars indicating one sample standard deviation. Dashed curves show power-law-with-floor fits.

Maximum	Training	OOS normalized RMSE		OOS raw RMSE
trajectories	trajectories	Mean	Median	Mean	Median
per system
1	20,471	$0.25605{\scriptscriptstyle\pm 0.00016}$	$0.22841{\scriptscriptstyle\pm 0.00029}$	$0.90963{\scriptscriptstyle\pm 0.00142}$	$0.23163{\scriptscriptstyle\pm 0.00017}$
2	40,845	$0.24897{\scriptscriptstyle\pm 0.00061}$	$0.22054{\scriptscriptstyle\pm 0.00072}$	$0.87511{\scriptscriptstyle\pm 0.00531}$	$0.22380{\scriptscriptstyle\pm 0.00081}$
4	81,239	$0.24292{\scriptscriptstyle\pm 0.00019}$	$0.21397{\scriptscriptstyle\pm 0.00026}$	$0.84550{\scriptscriptstyle\pm 0.00407}$	$0.21733{\scriptscriptstyle\pm 0.00036}$
8	160,304	$0.23925{\scriptscriptstyle\pm 0.00040}$	$0.20964{\scriptscriptstyle\pm 0.00021}$	$0.83187{\scriptscriptstyle\pm 0.00150}$	$0.21345{\scriptscriptstyle\pm 0.00036}$

Appendix

Table 11: Training-trajectory scaling across three seeds. Entries report the mean

{\scriptscriptstyle\pm}

sample standard deviation of each OOS trajectory statistic. All runs use the same 20,471 training systems and 20.11M-parameter model. Lower is better; the best average in each metric column is bolded.

Patch length $\bm{P}$	Patches / window	In-domain RMSE		OOS RMSE
		Raw	Normalized	Raw	Normalized
1	200	0.2669	0.22673	1.1814	0.22043
5	40	0.2278	0.20818	1.0674	0.20794
10	20	0.2281	0.20760	1.0548	0.20802
20	10	0.2317	0.20948	1.0660	0.20961
40	5	0.2329	0.20762	1.0825	0.21087

Appendix

Table 12: Patch-length sensitivity of FilterFM. In-domain values are trajectory-level means over the 280 test trajectories from the 14 main benchmark systems; OOS values are trajectory-level means over 8,207 held-out zero-shot trajectories. All runs use a single seed, predict the

0.1/0.5/0.9

quantiles, and use one prior-to-posterior state-reconstruction block. The shaded row is the

P=20

setting used in the main experiments. Lower is better.

Hyperparameter	Setting
Architecture and input representation
Window length $T$	200
Patch length $L_{p}$	20
Maximum observation / state dimension	100 / 100
Hidden width $d$	256
Self-attention heads	4

Appendix

Table 13: Architecture and training hyperparameters of the main FilterFM model.

Figure 11: Patch-level temporal attention from all four heads of the third patch-level block in the Table 1 checkpoint. Each system occupies one row and heads 1–4 occupy the columns. Within each heatmap, rows are query patches and columns are key patches; gray cells are causally masked. The marginal curves show the same unsmoothed observation channel in both directions, with alternating colors marking 20-step patches. All panels share a square-root color scale over weights from 0 to 1.

Figure 25

Figure 26

Figure 12: Coefficients

\alpha^{\ell,+}_{p,m,c}-0.5\alpha^{\ell,-}_{p,m,c}

for forming prototypes in Eq. ( 4 ), before the common

1/M

factor. Rows are prototype slots and columns are observation channels. Panels share a linear color scale: red indicates positive coefficients and blue negative coefficients.

Figure 13: Additional in-domain state-estimation examples on Tracking.

Figure 14: In-domain state estimation on Cubic-Sensoring100: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Panel of Figure 15 — Figure 15: Additional in-domain state-estimation examples on Pendulum.

Panel of Figure 17 — Figure 17: Additional in-domain state-estimation examples on Oscillator.

Figure 19: In-domain state estimation on Lorenz96: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 20: In-domain state estimation on VL20: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 21: In-domain state estimation on LorenzUV40: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 22: In-domain state estimation on Vanderpol64: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 23: In-domain state estimation on Rossler96: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 24: In-domain state estimation on Duffing64: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 25: In-domain state estimation on Burgers64: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 26: In-domain state estimation on Fhn64: six state channels from three held-out trajectories, with the same axis limits across methods in each row.

Figure 27: Zero-shot state estimation on two three-dimensional held-out systems. Each column shows one system, with all three state channels arranged from top to bottom. Black and red curves denote the ground truth and FilterFM posterior median, respectively; light-red bands indicate the $0.1$ – $0.9$ quantile intervals. All examples use the main checkpoint without fine-tuning.

Figure 28: Zero-shot state estimation on two additional three-dimensional held-out systems. Each column shows one system and all three state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 29: Zero-shot state estimation on two additional three-dimensional held-out systems. Each column shows one system and all three state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 30: Zero-shot state estimation on two additional three-dimensional held-out systems. Each column shows one system and all three state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 31: Zero-shot state estimation on two additional four-dimensional held-out systems. Each column shows one system and all four state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 32: Zero-shot state estimation on two additional four-dimensional held-out systems. Each column shows one system and all four state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 33: Zero-shot state estimation on two additional four-dimensional held-out systems. Each column shows one system and all four state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 34: Zero-shot state estimation on two additional four-dimensional held-out systems. Each column shows one system and all four state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 35: Zero-shot state estimation on two additional five-dimensional held-out systems. Each column shows one system and all five state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 36: Zero-shot state estimation on two additional five-dimensional held-out systems. Each column shows one system and all five state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 37: Zero-shot state estimation on two additional five-dimensional held-out systems. Each column shows one system and all five state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 38: Zero-shot state estimation on two additional five-dimensional held-out systems. Each column shows one system and all five state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 39: Zero-shot state estimation on two additional six-dimensional held-out systems. Each column shows one system and all six state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 40: Zero-shot state estimation on two additional six-dimensional held-out systems. Each column shows one system and all six state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 41: Zero-shot state estimation on two additional six-dimensional held-out systems. Each column shows one system and all six state channels, using the same checkpoint and plotting conventions as Figure 27 .

Figure 42: Zero-shot state estimation on two additional six-dimensional held-out systems. Each column shows one system and all six state channels, using the same checkpoint and plotting conventions as Figure 27 .

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