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PubblicazioneRepresentations of Atomic Environments and effects on Body-Ordering and Electrostatics on Machine-Learning Potentials(SISSA, 2026-10-07)Machine learning interatomic potentials (MLIPs) achieve DFT-level accuracy by replacing rigid functional forms with neural network architectures that allow for much more effective regression at the cost of less intuitive representations. This thesis explores how representations of atomic environments used by MLIPs behave in two regimes: at short range in how body-ordered expansions converge with respect to their basis functions, and at long range, in improving modeling the electronic response of systems sensitive to electrostatic interactions. We determined that Atomic Cluster Expansions (ACE) are only able to recover the DFT dimer curve if the self-interactions inherent in their efficient tensor product construction are filtered with a purification operator. Moreover, we highlight that the canonical many-body expansion has a non-monotonic and non-convergent scaling with higher body orders in trimers and systems with symmetric interactions. We also studied the ad hoc modeling of long range electrostatics through physically-motivated charge flow models, namely charge equilibration (QEq). We diagnosed how QEq-based treatments that parameterize through local atomic descriptors fail to accurately capture the correct dielectric response as diagnosed by the bare susceptibility $\bm{\chi}_0$, and extended the capabilities of our in-house code PANNA to include the split charge equilibration (SQE) model. Phonon dispersions from trained SQE models show signatures of finite longitudinal optical-transverse optical splitting in MgO where short range and charge equilibration counterparts fail to manifest, suggesting that the SQE model has the necessary ingredients to capture finite screening in ionic crystals. Both results show that physically-grounded constraints on MLIP representations recover behavior that unconstrained flexibility misses: recovering the canonical expansion through the purification methods as well as imposing restraints in charge flow through the bond topology in SQE.
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PubblicazioneReflection Positivity, Magnetic Lattice Laplacians and Emergent Phases in Lattice Fermionic Systems(SISSA, 2026-09-25)Emergent phenomena in interacting fermionic systems often arise as collective many-body effects whose formation can be understood by studying the interplay between geometry and symmetry. Understanding such phenomena from first principles remains one of the central challenges of Mathematical Physics, particularly in regimes where perturbative methods fail and non-trivial phases emerge through strongly correlated dynamics. This thesis investigates a class of lattice fermionic systems exhibiting \textit{Spontaneous Symmetry Breaking}, \textit{Flux ordering} and \textit{Topological Order}. Although the models considered belong to different physical frameworks, including Euclidean lattice field theories formulated in terms of Grassmann variables and Quantum Lattice Systems described by operators acting on Hilbert spaces, they share a common mathematical structure. In each case, geometric features encoded by lattice Dirac operators, magnetic fluxes, and discrete gauge fields interact with Reflection Positivity properties that allow for a rigorous non-perturbative analysis. A central role throughout the thesis is indeed played by \textit{Reflection Positivity} and one of its most striking consequences: the \textit{Chessboard Estimates}. These methods provide powerful tools for converting geometric information into quantitative bounds on energies, correlation functions, and phase stability. The first part of the thesis, based on \cite{fabbri2026chirallongrangeordereuclidean}, is devoted to lattice Gross–Neveu models in two dimensional Euclidean Spacetime. After a Hubbard–Stratonovich transformation, the fermionic theories are mapped to effective bosonic systems for which Reflection Positivity can be established. Combining Reflection Positivity methods with Peierls' Argument, we prove the existence of Chiral Long-Range Order for a class of lattice regularizations and obtain quantitative bounds relating the order parameter to the minimizers of the effective potential. The second part, based on \cite{GP}, focuses on fermions coupled to dynamical $\mathbb{Z}_2$-gauge fields. We prove the stability of the $\pi$-flux phase under gauging by showing that monopole excitations possess a strictly positive energy cost. These results imply the emergence of effective Dirac fermions at low energies and provide a rigorous characterization of the semimetallic phase of the model. Finally, we investigate the topological phase obtained by introducing a fermionic mass gap in the $\pi$-flux phase, following \cite{bachmann2026anyonspifluxphasefermionic}. We prove the existence of an almost four-fold degenerate low-energy sector on the torus, separated from the rest of the spectrum by a uniform gap, and construct quasi-local loop and string operators whose algebra exhibits the characteristic toric-code braiding phases. This provides a rigorous realization of topological order emerging from interacting fermions coupled to a dynamical gauge field. Taken together, these results illustrate how Reflection Positivity methods and spectral properties of the Magnetic Laplacian can be combined to obtain rigorous information on the phase diagram of interacting fermionic systems. They reveal a common mechanism underlying symmetry breaking, flux stabilization, and topological order across a variety of lattice models.
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PubblicazioneTheory of Learning in High-Dimensional Neural Networks: Generalization, Transfer Learning, and Neural Scaling Laws(SISSA, 2026-09-30)This thesis develops predictive theories for neural networks in high-dimensional settings, focusing on three problems of particular relevance to modern machine learning: generalization, transfer learning, and neural scaling laws. We first study the generalization error of two-layer neural networks with generic activation functions in a setting closer to empirical risk minimization, rather than restricting the analysis to an optimal statistical benchmark, and we characterize how the regularization strength and the ratio between the number of training samples and the number of trainable parameters shape the equilibrium solutions and the generalization error they achieve. For networks with Erf and ReLU activations, we identify phase transitions at which an equilibrium solution carrying non-trivial correlations with the ground-truth directions becomes optimal, and we show that the predicted generalization error is recovered by gradient-based learning algorithms. We then investigate transfer learning through Low-Rank Adaptation (LoRA), with particular emphasis on two of its essential ingredients, namely the pretrained weights and the rank of the low-rank update. Our analysis reveals a separation between two stages of the learning dynamics: an initial search phase, whose duration is primarily controlled by the pretrained weights and which ends once the network develops non-trivial correlations with the directions relevant to the downstream task, followed by a convergence phase in which the final generalization performance is controlled by the available adaptation rank. Finally, we study neural scaling laws in a controlled solvable setting and construct a model that reproduces the power-law decay of the error with model size observed empirically in large language models. We show that, when the model is optimally calibrated, the way the optimal norm of the learned directions varies with model width is by itself sufficient to generate the observed power law, and, beyond the exponent, we identify low-order geometric properties of the learned directions as the quantities controlling the scaling coefficient. This provides a mechanism for understanding how different learning algorithms can exhibit the same scaling exponent while reaching systematically different levels of performance through different scaling coefficients.
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PubblicazioneMolecular Dynamics Simulations of Flexible Biomolecules: Development of Enhanced Sampling Methods and Integration of Experimental Data(SISSA, 2026-09-28)Molecular dynamics (MD) simulations act as molecular microscopes, giving access to the position of every atom at every instant of time. This access comes at a price: the reliability of the result is limited by the accuracy of the underlying force field, and by the amount of conformational space that the simulation is actually able to explore. This thesis addresses both sides of this problem, by developing techniques that make the exploration of conformational space more efficient and by integrating experimental data into simulations, across chemically very different systems that share the property of being flexible. We first characterise the association of RNA with a fluid zwitterionic phospholipid bilayer using all-atom MD and well-tempered metadynamics, obtaining binding free energies and partition coefficients for systems ranging from single nucleosides to a 19-mer. In the absence of divalent cations, binding is driven by hydrogen bonding between nucleobases and lipid headgroups, with guanine emerging as the principal actor and the only nucleobase whose affinity increases systematically with sequence length. Base pairing competes with membrane binding, so that unstructured strands bind more strongly than folded ones of the same sequence, suggesting that disordered G-rich segments may be a hallmark of membrane-binding RNA molecules. We then address the cost of converging RNA ensembles. We introduce a protocol that optimises the Hamiltonian parametrisation of a replica-exchange ladder directly against a kinetic objective: from an existing set of HREX simulations, a Markov State Model of the generalised ensemble is built, reweighted to candidate parameters, and the round-trip times between the states whose populations one wants to converge are minimised. We show that the implied timescales of that model, which would appear to be the natural quantity to minimise, are the wrong objective, since they can be reduced by suppressing metastable states altogether. On adenosine, two rounds of optimisation yield a ladder sampling almost four times as many transitions as standard REST$2$; on tetranucleotides and hexamers the gains are modest but measurable, and transfer to sequences absent from the training set. Finally, we use EMMIVox to integrate experimental cryo-EM data into simulations of the E7--AP2 complex, in which a disordered viral oncoprotein binds a folded adaptor complex. Ensemble refinement reproduces the density substantially better than the original single structure, with the improvement concentrated in the voxels occupied by the disordered chains, and shows that Yxx$\Phi$ binding is preserved while the surrounding contacts are fuzzy and were likely over-counted by a single model. Taken together, these results illustrate a recurring point: for flexible systems, staying close to a single structure can be misleading, and it becomes necessary to study the ensemble, be it for small tetranucleotides or for fitting a cryo-EM density map of a large complex. Obtaining such ensembles is expensive, which is why part of this thesis is devoted to reducing that cost.
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PubblicazioneData-driven models for RNA: physical and statistical approaches to secondary structure, RNA–protein interactions, and siRNA potency(SISSA, 2026-09-28)RNA structures and interactions are often accessible only through indirect, noisy measurements. Principled models of how observations arise allow data-driven regression to infer not only measured labels but also latent biological quantities. This thesis develops such physical and statistical frameworks for RNA secondary structure, RNA--protein interactions, and siRNA potency. MERGE-RNA describes RNA structural ensembles by modelling the physics of dimethyl sulfate (DMS) probing. It learns transferable, interpretable parameters shared across molecules, probe concentrations, and replicates, while maximum entropy minimally adjusts thermodynamic populations to the data. Across systems and probe concentrations, inferred ensembles recapitulate measured DMS reactivity better than the traditional pseudo-free-energy model. On an adenine riboswitch, it recovers NMR-resolved conformations and ligand-induced rearrangement, with a structural midpoint matching the NMR-derived Kd; on a designed RNA, it resolves strand-displacement intermediate populations. For RNA-binding proteins, a generative model uses frozen foundation-model representations to predict eCLIP read counts and quantify fold-enrichment uncertainty under a shared input control. IP replicate tests support near-Poisson noise, while held-out SMInput genes validate the control rate. This separates reproducible signal from IP and control noise and estimates shared-control covariance. For Staufen-2 in HepG2 at 125-nucleotide resolution, counting noise accounts for 78% of fold-enrichment variance; control correction reduces replicate correlation from 0.54 to 0.20 and yields an honest prediction ceiling of 0.46. Though experiment- and resolution-dependent, this decomposition and correction apply whenever eCLIP replicates share a control. Silscore predicts siRNA potency from a position-resolved nearest-neighbour stacking-energy profile, a quadratic duplex-stability preference, and sparse identity terms selected from training residuals. Its 26-parameter main fit uses fewer than half the parameters of any trainable comparator. It has the highest Pearson correlation point estimate in three transfer evaluations, with a statistically supported advantage over every applicable comparator on the Katoh--Suzuki panel. When evaluated on a fixed test set after a change in the provenance of the training sequences, Silscore retains the highest correlation and shows the smallest decrease among fitted models. Its transferable, interpretable parameters formalise established rules within a principled data-driven framework, recovering an intermediate stability optimum, terminal asymmetry, and regional preferences for weaker seed and stronger central pairing. Together, these studies show how principled data-driven frameworks make latent quantities explicit and testable, with transfer and independent measurements probing whether they capture RNA biology.