Parametric Matrix Models: Equations as Data

Parametric Matrix Models Equations as Data Patrick Cook Facility for Rare Isotope Beams Michigan State University APS Division of Nuclear Physics October 2025 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation ␣pxi, yiq( Data y„fpxq ` ϵ Concepts y“ px`αqe´βx2sinpγxq Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation ␣pxi, yiq( Data y„fpxq ` ϵ Concepts y“ px`αqe´βx2sinpγxq Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation ␣pxi, yiq( Data y„fpxq ` ϵ Concepts y“ px`αqe´βx2sinpγxq Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation ␣pxi, yiq( Data y„fpxq ` ϵ Concepts y“ px`αqe´βx2sinpγxq Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation ␣pxi, yiq( Data y„fpxq ` ϵ Concepts y“ px`αqe´βx2sinpγxq Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation ␣pxi, yiq( Data y„fpxq ` ϵ Concepts y“ px`αqe´βx2sinpγxq Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation Data Concepts Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Motivation Data Concepts H|ψy “ E|ψy Equations Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Physics Informed Neural Networks (PINNs) L“ Data Loss ÿ iˇˇˇYi´ˆ Yiˇˇˇ 2 ` “Physics Loss” ÿ jˇˇˇMrˆ Yjsˇˇˇ 2 Trained (Online) Reduced Basis Methods (RBMs) snapshots Ñprojection P MRBM Pˆy» – | ˆ Y |fi fl P: m“PrMs Y«ˆ Y“P:ˆy Constructed (Offline) Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Physics Informed Neural Networks (PINNs) L“ Data Loss ÿ iˇˇˇYi´ˆ Yiˇˇˇ 2 ` “Physics Loss” ÿ jˇˇˇMrˆ Yjsˇˇˇ 2 Trained (Online) Reduced Basis Methods (RBMs) snapshots Ñprojection P MRBM Pˆy» – | ˆ Y |fi fl P: m“PrMs Y«ˆ Y“P:ˆy Constructed (Offline) Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Physics Informed Neural Networks (PINNs) L“ Data Loss ÿ iˇˇˇYi´ˆ Yiˇˇˇ 2 ` “Physics Loss” ÿ jˇˇˇMrˆ Yjsˇˇˇ 2 Trained (Online) Reduced Basis Methods (RBMs) snapshots Ñprojection P MRBM Pˆy» – | ˆ Y |fi fl P: m“PrMs Y«ˆ Y“P:ˆy Constructed (Offline) Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Physics Informed Neural Networks (PINNs) L“ Data Loss ÿ iˇˇˇYi´ˆ Yiˇˇˇ 2 ` “Physics Loss” ÿ jˇˇˇMrˆ Yjsˇˇˇ 2 Trained (Online) Reduced Basis Methods (RBMs) snapshots Ñprojection P MRBM Pˆy» – | ˆ Y |fi fl P: m“PrMs Y«ˆ Y“P:ˆy Constructed (Offline) Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) PMMs as Implicit Reduced Basis Methods Eigenvector Continuation Eigenvector Snapshots ÑP ` Parametric Hamiltonian With Known Operators Hpcq “ H0`cH1 Solve the projected problem hpcq “ PH0P: loomoon h0 `c PH1P: loomoon h1 Parametric Matrix Models Eigenvalue “Snapshots” ` Parametric Hamiltonian Form Hpcq“r?s ` cr?s Learn the projected problem hpcq “ h0`ch1 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) PMMs as Implicit Reduced Basis Methods Eigenvector Continuation Eigenvector Snapshots ÑP ` Parametric Hamiltonian With Known Operators Hpcq “ H0`cH1 Solve the projected problem hpcq “ PH0P: loomoon h0 `c PH1P: loomoon h1 Parametric Matrix Models Eigenvalue “Snapshots” ` Parametric Hamiltonian Form Hpcq“r?s ` cr?s Learn the projected problem hpcq “ h0`ch1 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) PMMs as Implicit Reduced Basis Methods Eigenvector Continuation Eigenvector Snapshots ÑP ` Parametric Hamiltonian With Known Operators Hpcq “ H0`cH1 Solve the projected problem hpcq “ PH0P: loomoon h0 `c PH1P: loomoon h1 Parametric Matrix Models Eigenvalue “Snapshots” ` Parametric Hamiltonian Form Hpcq“r?s ` cr?s Learn the projected problem hpcq “ h0`ch1 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) PMMs as Implicit Reduced Basis Methods Eigenvector Continuation Eigenvector Snapshots ÑP ` Parametric Hamiltonian With Known Operators Hpcq “ H0`cH1 Solve the projected problem hpcq “ PH0P: loomoon h0 `c PH1P: loomoon h1 Parametric Matrix Models Eigenvalue “Snapshots” ` Parametric Hamiltonian Form Hpcq“r?s ` cr?s Learn the projected problem hpcq “ h0`ch1 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) PMMs as Implicit Reduced Basis Methods Eigenvector Continuation Eigenvector Snapshots ÑP ` Parametric Hamiltonian With Known Operators Hpcq “ H0`cH1 Solve the projected problem hpcq “ PH0P: loomoon h0 `c PH1P: loomoon h1 Parametric Matrix Models Eigenvalue “Snapshots” ` Parametric Hamiltonian Form Hpcq“r?s ` cr?s Learn the projected problem hpcq “ h0`ch1 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) PMMs as Implicit Reduced Basis Methods Eigenvector Continuation Eigenvector Snapshots ÑP ` Parametric Hamiltonian With Known Operators Hpcq “ H0`cH1 Solve the projected problem hpcq “ PH0P: loomoon h0 `c PH1P: loomoon h1 Parametric Matrix Models Eigenvalue “Snapshots” ` Parametric Hamiltonian Form Hpcq“r?s ` cr?s Learn the projected problem hpcq “ h0`ch1 Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Background Parametric Matrix Models (PMMs) The Implicit Advantage Hpcq “ 1 2Nÿ i σx i`cσz i -2 2 c -1.00 -0.25 E 0 PMM EC True Train 2 256 N Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Recent Developments Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Nonlinear Operations Nonlinear Operations How do we construct an RBM for arbitrary nonlinear operations? Any non-affine elementwise operation on state vectors §|Ψ|2 §logp|Ψ|2q §expp|Ψ|2q §f:CnÑCn Key insight: the corresponding matrix operations are not elementwise fMpAq “ VAfpΛAqV: A ΨÑDpΨq “ » — – 0 Ψ 0 fi ffi fl Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Incomplete Equations How do we construct an RBM for incomplete equations? Dependence on some parameters is unknown or implicit §Basis-dependence §Finite-size effects §Temperature §Computational parameters Key insight: feature extraction with trainable universal approximators loooomoooon Feature Extraction loooomoooon Linear Model Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Incomplete Equations How do we construct an RBM for incomplete equations? Dependence on some parameters is unknown or implicit §Basis-dependence §Finite-size effects §Temperature §Computational parameters Key insight: feature extraction with trainable universal approximators loooomoooon Feature Extraction loooomoooon Linear Model Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Incomplete Equations How do we construct an RBM for incomplete equations? Dependence on some parameters is unknown or implicit §Basis-dependence §Finite-size effects §Temperature §Computational parameters Key insight: feature extraction with trainable universal approximators loooomoooon Feature Extraction loooomoooon Linear Model Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Incomplete Equations How do we construct an RBM for incomplete equations? Dependence on some parameters is unknown or implicit §Basis-dependence §Finite-size effects §Temperature §Computational parameters Key insight: feature extraction with trainable universal approximators loooomoooon Feature Extraction loooomoooon Linear Model Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Incomplete Equations How do we construct an RBM for incomplete equations? Dependence on some parameters is unknown or implicit §Basis-dependence §Finite-size effects §Temperature §Computational parameters Key insight: feature extraction with trainable universal approximators loooomoooon Feature Extraction loooomoooon Linear Model Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Hpc, αq “ H0pαq ` cH1pαq § § đ Hpc, αq “ ”Hp0q 0`fpαqHp1q 0ı`c”Hp0q 1`gpαqHp1q 1ı § § đ Mpc, αq “ ”Mp0q 0`fpαqMp1q 0ı looooooooooomooooooooooon M0pαq `c”Mp0q 1`gpαqMp1q 1ı loooooooooomoooooooooon M1pαq Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Hpc, αq “ H0pαq ` cH1pαq § § đ Hpc, αq “ ”Hp0q 0`fpαqHp1q 0ı`c”Hp0q 1`gpαqHp1q 1ı § § đ Mpc, αq “ ”Mp0q 0`fpαqMp1q 0ı looooooooooomooooooooooon M0pαq `c”Mp0q 1`gpαqMp1q 1ı loooooooooomoooooooooon M1pαq Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Incomplete Equations Hpc, αq “ H0pαq ` cH1pαq § § đ Hpc, αq “ ”Hp0q 0`fpαqHp1q 0ı`c”Hp0q 1`gpαqHp1q 1ı § § đ Mpc, αq “ ”Mp0q 0`fpαqMp1q 0ı looooooooooomooooooooooon M0pαq `c”Mp0q 1`gpαqMp1q 1ı loooooooooomoooooooooon M1pαq Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments System-Size Dependence System-Size Dependence All systems are just projections of a fundamental system, Hpc;Nq “ U: NHfundpcqUN “ Applying the implicit reduced basis, Mpc;Nq “ P:U: NpQQ:qHfundpcqpQQ:qUNP “ Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments System-Size Dependence System-Size Dependence All systems are just projections of a fundamental system, Hpc;Nq “ U: NHfundpcqUN “ Applying the implicit reduced basis, Mpc;Nq “ P:U: NpQQ:qHfundpcqpQQ:qUNP “ Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments System-Size Dependence Proof of Concept: Long Range Spin Model Hpc;Nq“´ N ÿ i‰j Jij `γxσx iσx j`γyσy iσy j˘`c N ÿ i σz i Mpc;Nq “ V:pNq`M0`cM1˘VpNq Given the ground state energy for various cand small N, can we predict the ground state energy for large N? Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments System-Size Dependence Proof of Concept: Long Range Spin Model Hpc;Nq“´ N ÿ i‰j Jij `γxσx iσx j`γyσy iσy j˘`c N ÿ i σz i Mpc;Nq “ V:pNq`M0`cM1˘VpNq Given the ground state energy for various cand small N, can we predict the ground state energy for large N? Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments System-Size Dependence Proof of Concept: Long Range Spin Model Hpc;Nq“´ N ÿ i‰j Jij `γxσx iσx j`γyσy iσy j˘`c N ÿ i σz i Mpc;Nq “ V:pNq`M0`cM1˘VpNq Given the ground state energy for various cand small N, can we predict the ground state energy for large N? Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments System-Size Dependence 3 14 N -1.2 -1.0 E 0 /N Train Val Test PMM 0 . 0 2 . 0 c Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Uncertainty Quantification How can we quantify uncertainty in PMM predictions? Bootstrapping with Ensemble Models §Train multiple models on resampled data §Use spread in predictions as uncertainty estimate Conformal Prediction §Distribution-free, finite-sample valid prediction intervals §Based on exchangeability of data Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Uncertainty Quantification How can we quantify uncertainty in PMM predictions? Bootstrapping with Ensemble Models §Train multiple models on resampled data §Use spread in predictions as uncertainty estimate Conformal Prediction §Distribution-free, finite-sample valid prediction intervals §Based on exchangeability of data Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Uncertainty Quantification How can we quantify uncertainty in PMM predictions? Bootstrapping with Ensemble Models §Train multiple models on resampled data §Use spread in predictions as uncertainty estimate Conformal Prediction §Distribution-free, finite-sample valid prediction intervals §Based on exchangeability of data Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Uncertainty Quantification How can we quantify uncertainty in PMM predictions? Bootstrapping with Ensemble Models §Train multiple models on resampled data §Use spread in predictions as uncertainty estimate Conformal Prediction §Distribution-free, finite-sample valid prediction intervals §Based on exchangeability of data Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Uncertainty Quantification How can we quantify uncertainty in PMM predictions? Bootstrapping with Ensemble Models §Train multiple models on resampled data §Use spread in predictions as uncertainty estimate Conformal Prediction §Distribution-free, finite-sample valid prediction intervals §Based on exchangeability of data Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Uncertainty Quantification How can we quantify uncertainty in PMM predictions? Bootstrapping with Ensemble Models §Train multiple models on resampled data §Use spread in predictions as uncertainty estimate Conformal Prediction §Distribution-free, finite-sample valid prediction intervals §Based on exchangeability of data Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Uncertainty Quantification Training Calibration Validation Testing §Uncertainty heuristic upxq §Calibration set scores si“ |yi´ˆyi|{upxiq §Take ˆqto be the rp1´αqpn`1qs nquantile of the scores §1´αconfidence intervals on predictions are now ˆy˘ˆqupxq Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together Putting It All Together N3LO IMSRG(2) Calculations of Symmetric Nuclear Matter Hpc;ρ, sq “ H0pρ, sq ` 4 ÿ i“1 ciHipρ, sq Can we calibrate the 4LECs cto reproduce empirical saturation properties? §Single calculation requires „10`hours on an H100 §Need to sample potentially thousands of LEC sets §„50% of flows diverge §Unknown dependence on flow parameter s §Basis dependence on density ρ §Uncertainty quantification Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together c s ρ αpc, ρq, βpc, ρq, γpc, ρq f1pρq,¨ ¨ ¨ M0`řiciMi αe´βs`γ Vpρq V:pρq“αe´βs`γ‰MpcqVpρqE0pc, ρ, sq Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Recent Developments Putting It All Together §Average test error „0.1MeV §1000 samples of LECs, „22 minutes on personal PC §ă1.5seconds per prediction §„104ˆ speedup 0.10 0.12 0.14 0.16 0.18 0.20 ρ sat (fm − 3) -20 -18 -16 -14 -12 -10 E sat /A (MeV) Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Summary Summary Parametric Matrix Models as Implicit Reduced Basis Methods hpcq “ h0`ch1 Incomplete Equations §Unknown dependence §Basis and system-size dependence Conformal Prediction §Uncertainty heuristic §Calibration dataset Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 Summary Summary Parametric Matrix Models as Implicit Reduced Basis Methods hpcq “ h0`ch1 Incomplete Equations §Unknown dependence §Basis and system-size dependence Conformal Prediction §Uncertainty heuristic §Calibration dataset Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025 References References Angelopoulos, Anastasios N. and Stephen Bates (2022). arXiv: 2107.07511 [cs.LG]. Cook, Patrick, Danny Jammooa, et al. (2025). In: Nature Communications 16.1, p. 5929. doi:10.1038/s41467-025-61362-4. De, Arinjoy, Patrick Cook, et al. (2025). In: Nature Communications 16.1, p. 7939. doi:10.1038/s41467-025-63398-y. Drischler, C., P. G. Giuliani, et al. (2024). In: Physical Review C 110.4. doi:10.1103/physrevc.110.044320. Goodfellow, Ian, Yoshua Bengio, et al. (2016). http://www.deeplearningbook.org. MIT Press. Patrick Cook (FRIB) parametric-matrix-models.github.io/pyPMM October 2025