cs.LGMar 16, 2026

Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems

Authors: Giacomo Albi, Alessandro Alla, Elisa Calzola

Organizations: Department of Computer Science, University of Verona, Verona, Italy · Department of Mathematics, La Sapienza University of Rome, Rome, Italy

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Abstract

We propose a data-driven framework to learn interaction kernels in stochastic multi-agent systems. Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data, without any a priori knowledge of the underlying interaction structure. Starting from a discrete stochastic binary-interaction model, we formulate the inverse problem as a sequence of sparse regression tasks in structured finite-dimensional spaces spanned by compactly supported basis functions, such as piecewise linear polynomials. In particular, we assume that pairwise interactions between agents are not directly observed and that only limited trajectory data are available. To address these challenges, we propose two complementary identification strategies. The first based on random-batch sampling, which compensates for latent interactions while preserving the statistical structure of the full dynamics in expectation. The second based on a mean-field approximation, where the empirical particle density reconstructed from the data defines a continuous nonlocal regression problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed framework, showing accurate reconstruction of both interaction and diffusion kernels even from partially observed. The method is validated on benchmark models, including bounded-confidence and attraction-repulsion dynamics, where the two proposed strategies achieve comparable levels of accuracy.

Figures & tables

Figure 1: Case of domain [0,2][0,2] , example of NbP=5N_{b}^{P}=5 piecewise linear basis functions on a uniform mesh (left) and on a Chebyshev mesh (right).
Figure 2: Test 1: Reconstructed kernel P(r)=(1+r2)−2P(r)=(1+r^{2})^{-2} (left) using NbP=10N^{P}_{b}=10 basis functions, reconstructed radial density D(r)=0.25/(1+r)2D(r)=0.25/(1+r)^{2} using NpD=8N^{D}_{p}=8 basis functions (center), comparison between the data density ff and the reconstructed f^\widehat{f} at the final time (right).
 F≡PF\equiv PF≡DF\equiv D
EF1E_{F}^{1}1.49⋅10−21.49\cdot 10^{-2}3.68⋅10−23.68\cdot 10^{-2}
EF∞E_{F}^{\infty}4.00⋅10−24.00\cdot 10^{-2}6.65⋅10−26.65\cdot 10^{-2}
Table 1: Test 1: Relative reconstruction errors for the interaction kernel PP and diffusion DD and for the empirical density f(⋅)f(\cdot) .
Figure 3: Test 1: Data used for the reconstruction of the kernel when Δ=4\Delta=4 and MP=25M_{P}=25 (left) and data used for the reconstruction of the diffusion with Δ=1\Delta=1 and MD=10M_{D}=10 .
Figure 4: Test 1. Reconstruction of the kernel P=χ(r<0.5)P=\chi(r<0.5) using 2121 basis functions, Δ=2\Delta=2 and MP=50M_{P}=50 using approach ( 34 ) (averaging) and ( 36 ) (best) in Alg. 1 and also the mean field approach described in Alg. 2 . (left), reconstruction of the density function D(x)D(x) using 1515 basis functions Δ=1\Delta=1 and MD=10M_{D}=10 (center), comparison between the data and the reconstructed density at the final time (right).
Averaging method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}
111.61⋅10−11.61\cdot 10^{-1}5.35⋅10−15.35\cdot 10^{-1}7.36⋅10−37.36\cdot 10^{-3}8.89⋅10−38.89\cdot 10^{-3}
221.12⋅10−11.12\cdot 10^{-1}5.12⋅10−15.12\cdot 10^{-1}6.68⋅10−36.68\cdot 10^{-3}9.20⋅10−39.20\cdot 10^{-3}
331.61⋅10−11.61\cdot 10^{-1}5.13⋅10−15.13\cdot 10^{-1}6.92⋅10−36.92\cdot 10^{-3}1.01⋅10−21.01\cdot 10^{-2}
Best result method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}
Table 2: Test 1. Errors for the identification of kernel P=χ(r<0.5)P=\chi(r<0.5) using NbP=21N_{b}^{P}=21 basis functions.
Figure 5: Test 2. Data used for the reconstruction of the kernel when ℓ=4\ell=4 and MP=25M_{P}=25 (left) and data used for the reconstruction of the diffusion with ℓ=1\ell=1 and MD=10M_{D}=10 .
Figure 6: Test 2. Reconstruction of kernel P(r)=((0.1+r)2−0.05(0.1+r)−2)/5P(r)=((0.1+r)^{2}-0.05(0.1+r)^{-2})/5 using 88 basis functions, ℓ=2\ell=2 and MP=50M_{P}=50 using approach ( 34 ) (averaging) and ( 36 ) (best) (left), with Np=1000N_{p}=1000 and K=10K=10 . Reconstruction of the density function D(x)D(x) using 1515 basis functions ℓ=1\ell=1 and M=10M=10 (center), comparison between the data and the reconstructed density at the final time (right).
Averaging method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}
111.41⋅10−11.41\cdot 10^{-1}1.65⋅10−11.65\cdot 10^{-1}4,98⋅10−34,98\cdot 10^{-3}4.77⋅10−34.77\cdot 10^{-3}
221.30⋅10−11.30\cdot 10^{-1}1.45⋅10−11.45\cdot 10^{-1}5.30⋅10−35.30\cdot 10^{-3}4.83⋅10−34.83\cdot 10^{-3}
332.09⋅10−12.09\cdot 10^{-1}2.48⋅10−12.48\cdot 10^{-1}5.14⋅10−35.14\cdot 10^{-3}5.12⋅10−35.12\cdot 10^{-3}
Best result method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}
Table 3: Test 2. Errors for the identification of kernel P(r)=((0.1+r)2−0.05(0.1+r)−2)/5P(r)=((0.1+r)^{2}-0.05(0.1+r)^{-2})/5 with NbP=8N_{b}^{P}=8 basis functions.
Figure 7: Test 3. Data used for the reconstruction of the kernel when ℓ=4\ell=4 and MP=25M_{P}=25 (left) and data used for the reconstruction of the diffusion in the mean field approach with ℓ=1\ell=1 and MD=20M_{D}=20 (right).
Figure 8: Test 3. Reconstruction of the kernel P(r)=(1+r2)−2P(r)=(1+r^{2})^{-2} using 1010 basis functions, S=3\mathscr{S}=3 , with approaches ( 34 ) (averaging) and ( 36 ) (best), Np=1000N_{p}=1000 , and K=10K=10 , compared with the mean-field method (left). Reconstruction of the density D(x)D(x) using 88 basis functions with S=2\mathscr{S}=2 for the random batch approach and ℓ=1\ell=1 , MD=10M_{D}=10 for the mean-field method (center). Comparison between the data and the reconstructed density at the final time (right).
Averaging method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}ED1E^{1}_{D}ED∞E^{\infty}_{D}
117.94⋅10−27.94\cdot 10^{-2}1.21⋅10−11.21\cdot 10^{-1}3.34⋅10−33.34\cdot 10^{-3}6.38⋅10−36.38\cdot 10^{-3}2.97⋅10−12.97\cdot 10^{-1}7.42⋅10−17.42\cdot 10^{-1}
227.53⋅10−27.53\cdot 10^{-2}1.06⋅10−11.06\cdot 10^{-1}3.35⋅10−33.35\cdot 10^{-3}7.37⋅10−37.37\cdot 10^{-3}3.04⋅10−13.04\cdot 10^{-1}7.42⋅10−17.42\cdot 10^{-1}
336.78⋅10−26.78\cdot 10^{-2}1.01⋅10−11.01\cdot 10^{-1}2.82⋅10−32.82\cdot 10^{-3}6.58⋅10−36.58\cdot 10^{-3}3.88⋅10−13.88\cdot 10^{-1}1.451.45
Best result method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}ED1E^{1}_{D}ED∞E^{\infty}_{D}
Table 4: Test 3. Errors for the identification of kernel P(r)=(1+r2)−2P(r)=(1+r^{2})^{-2} using 1010 basis functions and diffusion D(r)=(1+r)−2D(r)=(1+r)^{-2} using 88 basis functions.
Figure 9: Test 3. Reconstruction of kernel P(r)=((1+r2)−2P(r)=((1+r^{2})^{-2} using 1010 basis functions, ℓ=4\ell=4 and MP=25M_{P}=25 using approach ( 34 ) (averaging) and ( 36 ) (best), with Np=1000N_{p}=1000 and K=10K=10 , superposed with the mean field (left). Reconstruction of the density function D(x)D(x) using 88 basis functions ℓ=2\ell=2 and MD=50M_{D}=50 for the random batch approach and ℓ=1\ell=1 and MD=10M_{D}=10 for the mean field (center). Comparison between the data and the reconstructed density at the final time (right) using the reconstructed kernels and densities.
Averaging method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}ED1E^{1}_{D}ED∞E^{\infty}_{D}
117.20⋅10−27.20\cdot 10^{-2}1.21⋅10−11.21\cdot 10^{-1}1.78⋅10−31.78\cdot 10^{-3}3.58⋅10−33.58\cdot 10^{-3}7.68⋅10−27.68\cdot 10^{-2}3.33⋅10−13.33\cdot 10^{-1}
228.77⋅10−28.77\cdot 10^{-2}1.49⋅10−11.49\cdot 10^{-1}2.18⋅10−32.18\cdot 10^{-3}5.86⋅10−35.86\cdot 10^{-3}7.66⋅10−27.66\cdot 10^{-2}3.33⋅10−13.33\cdot 10^{-1}
336.70⋅10−26.70\cdot 10^{-2}1.26⋅10−11.26\cdot 10^{-1}2.24⋅10−32.24\cdot 10^{-3}5.29⋅10−35.29\cdot 10^{-3}7.59⋅10−27.59\cdot 10^{-2}3.33⋅10−13.33\cdot 10^{-1}
Best result method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}EfaveE^{\textrm{ave}}_{f}EffinE^{\textrm{fin}}_{f}ED1E^{1}_{D}ED∞E^{\infty}_{D}
Table 5: Test 3. Errors for the identification of kernel P(r)=(1+r2)−2P(r)=(1+r^{2})^{-2} using 1010 basis functions and diffusion D~(r)=(1+r)−2\tilde{D}(r)=(1+r)^{-2} using 88 basis functions.
Figure 10: Test 4. (Test 2D) Reconstruction of kernel P=(−(0.1+r)−1.15+r2)/2P=(-(0.1+r)^{-1.15}+r^{2})/2 using 1010 basis functions, ℓ=2\ell=2 and MP=50M_{P}=50 (left), reconstruction of the density function D1(x1)D_{1}(x_{1}) using 1515 basis functions ℓ=1\ell=1 and MD1=20M_{D_{1}}=20 (center), reconstruction of the density function D2(x2)D_{2}(x_{2}) using 1515 basis functions ℓ=1\ell=1 and MD2=20M_{D_{2}}=20 (right).
Figure 11: Test 4. Evolution of the reconstructed density (left) and trend of the normalized L1L^{1} errors over time for the different reconstruction strategies (right), setting S=3\mathscr{S}=3 .
Averaging method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}Ef1E^{1}_{f}EfT1E^{1}_{f_{T}}
221.83⋅10−11.83\cdot 10^{-1}2.92⋅10−12.92\cdot 10^{-1}1.21⋅10−11.21\cdot 10^{-1}1.42⋅10−11.42\cdot 10^{-1}
332.30⋅10−12.30\cdot 10^{-1}2.52⋅10−12.52\cdot 10^{-1}1.19⋅10−11.19\cdot 10^{-1}1.41⋅10−11.41\cdot 10^{-1}
Best result method (Alg. 1 )
S\mathscr{S}EP1E^{1}_{P}EP∞E^{\infty}_{P}Ef1E^{1}_{f}EfT1E^{1}_{f_{T}}
114.06⋅10−14.06\cdot 10^{-1}3.19⋅10−13.19\cdot 10^{-1}8.83⋅10−28.83\cdot 10^{-2}9.22⋅10−29.22\cdot 10^{-2}
Table 6: Test 4. Errors for the identification of kernel P=(−(0.1+r)−1.15+r2)/2P=(-(0.1+r)^{-1.15}+r^{2})/2 using 1010 basis functions. The errors on the diffusion along the first variable are ED11=4.40⋅10−2E^{1}_{D_{1}}=4.40\cdot 10^{-2} and ED1∞=1.11⋅10−2E^{\infty}_{D_{1}}=1.11\cdot 10^{-2} , while the errors on the diffusion along the second variable are ED21=3.47⋅10−2E^{1}_{D_{2}}=3.47\cdot 10^{-2} and ED1∞=9.82⋅10−2E^{\infty}_{D_{1}}=9.82\cdot 10^{-2} .
Figure 12: Test 5. (Test 2D) Reconstruction of kernel P=χ(r<1)P=\chi(r<1) using NbP=21N_{b}^{P}=21 , ℓ=4\ell=4 and MP=25M_{P}=25 (left), reconstruction of the density function D~(r)\tilde{D}(r) using NbD=10N_{b}^{D}=10 , ℓ=4\ell=4 and MD2=25M_{D_{2}}=25 (right).
Figure 13: Test 5. Evolution of the reconstructed density (left) and trend of the normalized L1L^{1} errors over time for the different reconstruction strategies (right), setting S=3\mathscr{S}=3 .
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