SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 14: Results of regular time series forecasting. We set the input length as 36 for ILI and 96 for the others. Avg. stands for Average. Bold values indicate the best performance and underlined values indicate the second-best performance (lower the better).
Models Metrics	ContiFormer	TimesNet[52]	DLinear[56]	FEDformer[62]	Autoformer[53]	Transformer[51]	Informer[61]	Reformer[31]
MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Exchange	96	0.105	0.232	0.107	0.238	0.088	0.217	0.153	0.284	0.153	0.283	0.891	0.743	1.348	0.966	1.208	0.961
192	0.215	0.339	0.201	0.327	0.175	0.313	0.264	0.375	0.327	0.413	1.338	0.955	1.296	0.941	1.329	1.001
336	0.336	0.427	0.354	0.435	0.310	0.421	0.437	0.485	0.555	0.560	1.553	1.056	1.641	1.072	3.023	1.427
540	0.698	0.598	0.614	0.582	0.534	0.555	0.710	0.639	0.724	0.648	1.515	1.042	2.430	1.284	1.801	1.159
Avg.	0.339	0.399	0.320	0.396	0.277	0.377	0.392	0.446	0.440	0.477	1.325	0.949	1.679	1.066	1.841	1.137
ETTm2	24	0.126	0.226	0.104	0.200	0.109	0.213	0.149	0.253	0.157	0.262	0.282	0.397	0.391	0.475	0.405	0.499
48	0.175	0.276	0.143	0.238	0.145	0.249	0.172	0.268	0.177	0.276	0.501	0.535	0.420	0.484	0.541	0.569
168	0.272	0.336	0.255	0.308	0.262	0.345	0.255	0.318	0.262	0.324	0.824	0.730	1.068	0.869	1.149	0.846
336	0.331	0.361	0.352	0.359	0.366	0.414	0.326	0.361	0.328	0.365	1.149	0.871	1.379	0.983	2.258	1.153
Avg.	0.226	0.300	0.214	0.276	0.220	0.305	0.226	0.300	0.231	0.307	0.689	0.633	0.815	0.703	1.088	0.767
ETDb2	24	0.237	0.327	0.196	0.281	0.179	0.277	0.262	0.340	0.266	0.343	0.946	0.800	1.408	1.028	0.762	0.703
48	0.287	0.355	0.244	0.317	0.247	0.329	0.291	0.354	0.294	0.358	1.009	0.826	1.642	1.100	1.101	0.856
168	0.417	0.426	0.397	0.409	0.430	0.448	0.403	0.420	0.403	0.419	3.014	1.483	3.261	1.511	2.076	1.144
336	0.484	0.471	0.446	0.449	0.577	0.532	0.448	0.458	0.445	0.456	3.357	1.606	3.155	1.481	2.633	1.297
Avg.	0.356	0.395	0.321	0.364	0.358	0.396	0.351	0.393	0.352	0.394	2.081	1.179	2.366	1.280	1.643	1.000
ILI	24	2.391	1.004	2.925	1.061	4.412	1.576	4.979	1.668	4.334	1.527	4.789	1.431	5.718	1.623	4.678	1.455
36	2.673	1.006	3.145	1.127	4.314	1.529	4.812	1.628	4.084	1.438	4.995	1.474	5.353	1.564	4.801	1.490
48	2.536	1.039	2.716	1.027	3.910	1.424	4.112	1.465	3.870	1.408	5.283	1.545	5.120	1.537	4.946	1.510
60	2.930	1.117	2.711	1.049	4.114	1.446	3.965	1.428	3.895	1.411	5.454	1.578	5.556	1.613	5.134	1.545
Avg.	2.632	1.042	2.874	1.066	4.188	1.493	4.467	1.547	4.046	1.446	5.130	1.507	5.437	1.584	4.890	1.500
Weather	96	0.170	0.218	0.171	0.221	0.194	0.250	0.254	0.323	0.253	0.317	0.579	0.539	0.599	0.541	0.940	0.675
192	0.229	0.271	0.230	0.269	0.237	0.297	0.287	0.344	0.293	0.347	0.738	0.613	0.459	0.453	1.061	0.735
336	0.288	0.317	0.289	0.302	0.282	0.334	0.336	0.374	0.346	0.380	0.464	0.466	0.575	0.495	1.201	0.802
540	0.341	0.355	0.345	0.348	0.325	0.365	0.384	0.400	0.379	0.394	0.909	0.700	0.862	0.625	0.958	0.702
Avg.	0.257	0.290	0.259	0.285	0.260	0.311	0.315	0.360	0.318	0.359	0.672	0.579	0.624	0.529	1.040	0.729
measuring 24 by 24 pixels, and is afflicted by correlated noise. These images are sampled at irregular intervals from a continuous trajectory spanning a duration of 100 units, while the targets correspond to the sine and cosine values of the pendulum's angle. It's important to note that the pendulum follows a nonlinear dynamical system, with the velocity remaining unobserved. Table 15 summarizes the results of this experiment. ContiFormer outperforms CRU and ODE-RNN.
Table 15: Regression MSE  $\times  {10}^{-3}$  (mean  $\pm$  std) on pendulum regression task.
Model	Regression MSE (×10-3)
mTAN [48]	65.64 (4.05)
ODE-RNN [43]	7.26 (0.41)
CRU [45]	4.63 (1.07)
S5 [49]	3.41 (0.27)
ContiFormer	4.21 (0.24)
D Ablation Study and Parameter Study
D.1 Effects of Different Attention Mechanism
We study the impact of different attention mechanisms to show the significance of both continuous-evolving attention scores and continuous value functions. We claim that both of these two continuous designs attribute to the modeling of complex dynamic systems. We conduct the experiment on 6 datasets for event prediction task. As shown in Table. 16, ContiFormer outperforms both of the two variants, which shows that jointly modeling the dynamic relationship between input observations and meanwhile capturing the dynamic change of the observations is critical for modeling complex event sequences in reality.
D.2 Effects of Different Numeric Approximation Methods
As discussed in Appendix A.2, linear interpolation and Gauss-Legendre Quadrature are two methods for approximating an integral in a close interval  $[-1, 1]$ . We study the impact of these two methods and set  $P$ , i.e., the number of intermediate steps for integral approximation, in the Gauss-Legendre quadrature from  $\{2, 3, 4, 5\}$ . We conducted the experiment on 6 datasets for event prediction task.
Table 16: Effects of different attention mechanism design on event prediction task. Attn. stands for Attention.  $\checkmark$  refers to a continuous version of attention/value, while  $\times$  refers to a static version. It is important to note that when both attention and value are static, the model is equivalent to Transformer. Conversely, when both are continuous, the model is referred to as ContiFormer.
Dataset	Synthetic	Neonate	Traffic
Attn.	Value	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
X	X	-0.589	0.841	0.205	-2.702	9.471	0.569	0.818	0.332
X	✓	-0.576	0.842	0.202	-2.634	9.249	0.598	0.819	0.327
✓	X	-0.563	0.842	0.200	-2.590	9.267	0.544	0.814	0.329
✓	✓	-0.535	0.842	0.192	-2.550	9.233	0.635	0.822	0.328
Dataset	MIMIC	BookOrder	StackOverflow
Attn.	Value	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
X	X	-1.137	0.834	0.843	-0.302	0.622	3.688	-2.354	0.468	0.951
X	✓	-1.147	0.832	0.837	-0.312	0.627	3.690	-2.354	0.468	0.950
✓	X	-1.137	0.837	0.838	-0.272	0.629	3.632	-2.337	0.472	0.948
✓	✓	-1.135	0.836	0.837	-0.270	0.628	3.614	-2.334	0.472	0.948
Table 17: Effects of difference numerical approximation methods on event prediction task. (default to linear approximation instead of the Gauss-Legendre quadrature method).
Dataset P	Synthetic	Neonate	Traffic
LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
Approxiamation Method = Linear
-	-0.535	0.842	0.192	-2.550	9.233	0.635	0.822	0.328
Approxiamation Method = Gauss-Legendre Quadrature
2	-0.545	0.842	0.198	-2.571	9.235	0.599	0.817	0.330
3	-0.531	0.842	0.194	-2.594	9.236	0.610	0.820	0.328
4	-0.556	0.842	0.199	-2.590	9.253	0.589	0.819	0.328
5	-0.550	0.842	0.193	-2.602	9.258	0.601	0.820	0.328
Dataset P	MIMIC	BookOrder	StackOverflow
LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
Approxiamation Method = Linear
-	-1.135	0.836	0.837	-0.270	0.628	3.614	-2.332	0.473	0.948
Approxiamation Method = Gauss-Legendre Quadrature
2	-1.144	0.834	0.840	-0.291	0.627	3.653	-2.337	0.472	0.948
3	-1.142	0.837	0.839	-0.294	0.627	3.667	-2.337	0.471	0.948
4	-1.143	0.837	0.839	-0.287	0.627	3.647	-2.340	0.471	0.948
5	-1.132	0.834	0.838	-0.290	0.628	3.563	-2.338	0.471	0.948
The experimental results can be found in Table 17. As shown in the table, our model exhibits low sensitivity to the choice of approximation methods.
D.3 Effects of Tolerance Errors in ODESolver
ODE solvers play a crucial role in ensuring that the model's output remains within a specified tolerance of the true solution [9]. By adjusting this tolerance, we can observe changes in the behavior of the neural network. Given our use of a fixed-step ODE solver, we sought to investigate the impact of the step size on prediction performance throughout the experiment. Specifically, we explored different step sizes from the set  $\{0.5, 0.1, 0.05, 0.01, 0.005\}$ . A smaller step size corresponds to a greater number of forward passes required to solve the ordinary differential equations, resulting in more accurate solutions at the expense of increased computational time. We conducted extensive experiments on 6 datasets for the event prediction task. The experimental results are presented in Table 18. The table reveals that our model demonstrates low sensitivity to the tolerance error. Therefore, we set the tolerance error to 0.1 in order to make a desirable balance between computational time and prediction accuracy. We explain the phenomena in two aspects. First, our framework, similar to the
Table 18: Effects of tolerance errors in ODESolver on event prediction task (default to 0.5 for Neonate dataset and 0.1 for other datasets). (The use of a double dash  $(-)$  indicates that the experiment requires an excessively long duration of time.)
Dataset P	Synthetic	Neonate	Traffic
LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
5e-1	-0.559	0.841	0.196	-2.550	9.233	0.614	0.820	0.328
1e-1	-0.535	0.842	0.192	-2.561	9.229	0.635	0.822	0.328
5e-2	-0.549	0.842	0.195	-2.568	9.236	0.620	0.819	0.329
1e-2	-0.539	0.842	0.195	-2.573	9.233	0.619	0.820	0.329
5e-3	-0.549	0.842	0.199	-2.559	9.238	0.613	0.819	0.329
Dataset Stepsize	MIMIC	BookOrder	StackOverflow
LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
5e-1	-1.138	0.837	0.837	-0.268	0.629	3.619	-2.333	0.473	0.948
1e-1	-1.135	0.836	0.837	-0.270	0.628	3.614	-2.332	0.473	0.948
5e-2	-1.137	0.835	0.837	-0.263	0.629	3.609	-2.332	0.473	0.948
1e-2	-1.143	0.836	0.838	-0.263	0.629	3.612	-	-	-
5e-3	-1.147	0.837	0.838	-	-	-	-	-	-
Transformer architecture, circumvents cumulative errors by eliminating the necessity of sequentially passing neural networks in a regressive manner. Second, since the output from the attention module is actually a weighted sum of the tokens, the total variance is lowered. For instance, assume that  $X_{1}, X_{2}, \ldots, X_{N} \sim \mathcal{N}(\mu, \sigma^{2})$ , then the total variance of the mean value  $\bar{x} = \frac{1}{N} (X_{1} + X_{2} + \ldots + X_{N})$  is  $\frac{\sigma^2}{N}$ , and the variance is significantly lowered given the large. Overall, our model is not sensitive to tolerance error, which makes our model robust to different scenarios.
D.4 Effects of Vector Fields in ODE
Neural ODEs are controlled by a vector field that defines the change of  $x$  over time, i.e.,
$$
\frac {\mathrm {d} \mathbf {x}}{\mathrm {d} t} = f (\mathbf {x}, t). \tag {56}
$$
where  $f:\mathbb{R}^{d + 1}\to \mathbb{R}^d$ . We investigate two types of  $f$  to implement Eq. (1).
Specifically, we denote Concat as
$$
f (\mathbf {x}, t) = \operatorname {A c t f n} \left(\operatorname {L i n e a r} ^ {d, d} \left(\operatorname {L i n e a r} ^ {d, d} (\mathbf {x}) + \operatorname {L i n e a r} ^ {1, d} (t)\right)\right), \tag {57}
$$
and ConcatNorm as
$$
f (\mathbf {x}, t) = \operatorname {A c t f n} \left(\operatorname {L N} \left(\operatorname {L i n e a r} ^ {d, d} \left(\operatorname {L i n e a r} ^ {d, d} (\mathbf {x}) + \operatorname {L i n e a r} ^ {1, d} (t)\right)\right)\right), \tag {58}
$$
where  $\operatorname{Actfn}(\cdot)$  is either tanh or sigmoid activation function,  $\operatorname{Linear}^{a,b}(\cdot): \mathbb{R}^a \to \mathbb{R}^b$ : is a linear transformer from dimension  $a$  to dimension  $b$ , LN denotes the layer normalization. We conduct the experiment on 6 datasets for event prediction task. The experimental results are shown in Table 19. As shown in the table, incorporating normalization before the activation function leads to improved predictive performance.
D.5 Effects of Dropout Rate
Dropout in the Transformer can prevent overfitting and boost performance. However, with the increment of the dropout rate, we output of ContiFormer tends to be less continuous, leading to a sub-optimal solution for tasks that prefer a continuous-time output. To this end, we conduct experiments on both modeling continuous-time function and event prediction tasks. As for the task of modeling continuous-time function, it is desirable to have a dynamic system where outputs are "almost" continuous and smooth. As shown in Figure 5, the performance of both Transformer and ContiFormer drops significantly as the dropout rate increases. For prediction tasks, the dynamic system can be shaky, which reflects noise or other uncontrolled influences on the system. The experimental results on event prediction can be found in Table 20. As shown in the table, the dropout rate has low sensitivity to the prediction results.
Table 19: Effects of different vector field designs on event prediction task. Attn. stands for Attention.  
✓ refers to ConcatNorm, while × refers to Concat.
Dataset	Synthetic	Neonate	Traffic
Norm	Actfn	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
×	tanh	-0.552	0.842	0.198	-2.550	9.233	0.639	0.820	0.329
×	sigmoid	-0.561	0.842	0.201	-2.602	9.263	0.635	0.822	0.328
✓	tanh	-0.556	0.842	0.199	-2.584	9.242	0.628	0.820	0.328
✓	sigmoid	-0.535	0.842	0.192	-2.601	9.232	0.616	0.819	0.329
Dataset	MIMIC	BookOrder	StackOverflow
Norm	Actfn	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
×	tanh	-1.162	0.831	0.837	-0.274	0.627	3.615	-2.337	0.472	0.948
×	sigmoid	-1.144	0.836	0.838	-0.270	0.628	3.614	-2.334	0.472	0.948
✓	tanh	-1.157	0.833	0.840	-0.307	0.622	3.681	-2.335	0.473	0.947
✓	sigmoid	-1.135	0.836	0.837	-0.302	0.625	3.689	-2.332	0.473	0.948
Figure 5: Interpolation results on 2D spiral under different dropout rates.
D.6 Effects of Time Normalization
As shown in Eq. (3) and Eq. (5), due to the nature of sequence data, it is common to encounter abnormal points, such as events with significantly large time differences. To avoid numeric instability during training, we divide the integrated solution by the time difference. To examine the effect of normalization, we conduct experiments on 6 datasets for event prediction task. The experimental results can be found in Table 21, which shows that the normalization is significant on most datasets, including Synthetic, Neonate, MIMIC and StackOverflow datasets.
E Time Cost v.s. Memory Cost v.s. Accuracy
Figure 6: Time Cost v.s. Memory Cost v.s. Accuracy.
Utilizing continuous-time modeling in ContiFormer often results in substantial time and GPU memory overhead. We have conducted a comparative analysis of ContiFormer's actual time costs in comparison to Transformer-based models (TST), ODE-based models (ODE-RNN, Neural CDE), and SSM-based models (S5). Throughout our analysis, we employ an RK4 solver for ODE-based models and our model, setting the step size to 0.1 and 0.01. It's worth noting that, for ODE-RNN and other
Table 20: Effects of dropout rate in ContiFormer on event prediction task (default to 0.1).
Dataset Dropout	Synthetic	Neonate	Traffic
LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
0	-0.557	0.842	0.191	-2.588	9.215	0.633	0.824	0.328
0.1	-0.535	0.842	0.192	-2.550	9.233	0.635	0.822	0.328
0.3	-0.644	0.842	0.223	-2.583	9.242	0.577	0.814	0.332
0.5	-0.696	0.841	0.230	-2.579	9.280	0.525	0.813	0.331
Dataset Dropout	MIMIC	BookOrder	StackOverflow
LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
0	-1.193	0.825	0.839	-0.268	0.629	3.616	-2.338	0.472	0.949
0.1	-1.135	0.836	0.837	-0.270	0.628	3.614	-2.332	0.473	0.948
0.3	-1.142	0.833	0.836	-0.266	0.630	3.608	-2.336	0.472	0.948
0.5	-1.149	0.834	0.844	-0.270	0.629	3.624	-2.342	0.471	0.948
Table 21: Effects of normalization on event prediction task. Attn. stands for Attention.  $\checkmark$  refers to normalizing the attention/value as shown in Eq. (3)/Eq. (5), while  $\times$  refers to removing the normalization, i.e.  $t - t_{i}$ . Note that ContiFormer use both normalization for attention and values.
Dataset	Synthetic	Neonate	Traffic
Attn.	Value	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
X	X	-0.618	0.842	0.215	-2.612	9.277	0.721	0.822	0.329
X	✓	-0.601	0.841	0.206	-2.576	9.240	0.536	0.814	0.336
✓	X	-0.584	0.842	0.205	-2.601	9.272	0.576	0.818	0.327
✓	✓	-0.535	0.842	0.192	-2.550	9.233	0.635	0.822	0.328
Dataset	MIMIC	BookOrder	StackOverflow
Attn.	Value	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)	LL (↑)	ACC (↑)	RMSE (↓)
X	X	-1.167	0.826	0.848	-0.311	0.627	3.710	-2.346	0.470	0.948
X	✓	-1.149	0.833	0.840	-0.269	0.630	3.634	-2.341	0.471	0.948
✓	X	-1.172	0.826	0.844	-0.277	0.629	3.654	-2.336	0.472	0.948
✓	✓	-1.135	0.836	0.837	-0.270	0.628	3.614	-2.334	0.472	0.948
ODE-based models, the input times have an impact on the number of forward passes when using a fixed-step ODE solver, thus, affecting the time costs.
The left figure in Figure 6 illustrates the influence of input times concerning input length. For the model with normalization (a.k.a., w/ norm), the input times consistently range from [0, 1], regardless of the input length. Conversely, for the model without normalization (a.k.a., w/o norm), the input times span from  $[0, L]$ , where  $L$  represents the input length.
As depicted in the middle figure in Figure 6, our model performs comparably to ODE-RNN (w/ norm) when the input length is up to 100, but becomes approximately four times slower as the input length extends to 1000. The right figure in Figure 6 displays GPU memory usage, revealing a significantly higher memory cost associated with our model.
Table 22 provides a more comprehensive overview of different models on UEA datasets. It is evident that our model strikes a better balance between accuracy and time cost. Specifically, our model achieves the best performance on 3 out of 6 selected datasets while incurring only twice the time cost compared to S5.
Table 22: Time Cost v.s. Accuracy for different models on UEA classification datasets (0% dropped). For all ODE-based models and our model, we adopt the RK4 solver and step size refers to the amount of time increment for the RK4 solver.  $\uparrow (\downarrow)$  indicates the higher (lower) the better. Note that since the results are aggregated over 3 random seeds, achieving the same results doesn't indicate the same outcome for each random seed.
Dataset
Model	Step Size	BasicMotions	JapaneseVowels	Libras
Time Cost (↓)	Accuracy (↑)	Time Cost (↓)	Accuracy (↑)	Time Cost (↓)	Accuracy (↑)
TST	-	0.56×	0.975	0.10×	0.986	0.36×	0.843
S5	-	1×	0.958	1×	0.926	1×	0.655
ODE-RNN (w/o norm)	0.1	3.06×	0.917	0.79×	0.952	2.67×	0.602
0.01	18.94×	0.925	5.26×	0.955	16.29×	0.611
ODE-RNN (w/ norm)	0.1	1.46×	1.000	0.32×	0.980	1.24×	0.637
0.01	1.59×	1.000	0.47×	0.980	1.59×	0.619
Neural CDE	0.1	4.47×	0.958	1.16×	0.932	3.87×	0.763
0.01	34.26×	0.958	9.57×	0.942	28.48×	0.744
ContiFormer	0.1	2.89×	0.975	0.49×	0.990	2.26×	0.870
0.01	13.75×	0.975	1.87×	0.991	8.74×	0.870
Dataset
Model	NATOPS	PEMS-SF	RacketSports
Step Size	Time Cost (↓)	Accuracy (↑)	Time Cost (↓)	Accuracy (↑)	Time Cost (↓)	Accuracy (↑)
TST	-	0.32×	0.963	0.15×	0.890	0.14×	0.826
S5	-	1×	0.839	1×	0.896	1×	0.772
ODE-RNN (w/o norm)	0.1	2.67×	0.909	4×	0.778	2.47×	0.827
0.01	16.52×	0.907	28.45×	0.761	14.65×	0.796
ODE-RNN (w/ norm)	0.1	1.28×	0.898	1.47×	0.775	1.15×	0.785
0.01	1.45×	0.898	1.44×	0.775	1.57×	0.787
Neural CDE	0.1	3.83×	0.789	6.04×	0.763	3.48×	0.743
0.01	29.24×	0.787	50.36×	0.763	25.04×	0.748
ContiFormer	0.1	2.32×	0.935	5.45×	0.823	1.87×	0.836
0.01	9.76×	0.935	37.08×	0.823	5.88×	0.836
F Broader Impact
The proposed model, ContiFormer, presents several potential impacts within the field of machine learning and irregular time series analysis. These include:
- Enhanced Modeling Capability: ContiFormer extends the Transformer architecture to effectively capture complex continuous-time dynamic systems. By leveraging continuous-time modeling techniques, the model demonstrates improved performance in handling irregular time series data, offering a more flexible and powerful modeling ability.
- Practical Applications: Our proposed method achieves state-of-the-art performance in a variety of tasks and real-life datasets, which makes it more promising to tackle real-world applications. The ability to model continuous-time dynamics offers valuable insights into time-evolving systems, facilitating improved decision-making and predictive capabilities in diverse domains.