LEARNING EFFICIENT HYBRID PARTICLE-CONTINUUM REPRESENTATIONS OF NON-EQUILIBRIUM N-BODY SYSTEMS

Abstract

An important class of multi-scale, non-equilibrium, N-body physical systems deals with an interplay between particle and continuum phenomena. These include hypersonic flow and plasma dynamics, materials science, and astrophysics. Hybrid solvers that combine particle and continuum representations could provide an efficient framework to model these systems. However, the coupling between these two representations has been a key challenge, which is often limited to inaccurate or incomplete prescriptions. In this work, we introduce a method for Learning Hybrid Particle-Continuum (LHPC) models from the data of first-principles particle simulations. LHPC analyzes the local velocity-space particle distribution function and separates it into near-equilibrium (thermal) and far-from-equilibrium (non-thermal) components. The most computationally-intensive particle solver is used to advance the non-thermal particles, whereas a neural network solver is used to efficiently advance the thermal component using a continuum representation. Most importantly, an additional neural network learns the particle-continuum coupling: the dynamical exchange of mass, momentum, and energy between the particle and continuum representations. Training of the different neural network components is done in an integrated manner to ensure global consistency and stability of the LHPC model. We demonstrate our method in an intense laser-plasma interaction problem involving highly nonlinear, far-from-equilibrium dynamics associated with the coupling between electromagnetic fields and multiple particle species. More efficient modeling of these interactions is critical for the design and optimization of compact accelerators for material science and medical applications. Our method achieves an important balance between accuracy and speed: LHPC is 8× faster than a classical particle solver and achieves up to 6.8-fold reduction of long-term prediction error for key quantities of interest compared to deep-learning baselines using uniform representations.

1. INTRODUCTION

The dynamics of physical systems is often nonlinear and involves the competition of different processes across a wide range of spatial and temporal scales. This gives rise to local non-equilibrium conditions (in the thermodynamic sense) that results in the failure of common numerical approaches. While continuum models (e.g., based on fluid equations) can provide accurate and computationally efficient descriptions of near-equilibrium systems at large scales, they break down when significant departures from local equilibrium are encountered (often at small scales) and give rise to important N-body phenomena. Kinetic (e.g., particle-based) numerical methods can accurately describe these non-equilibrium phenomena but are very computationally intensive, limiting their practical application to small scales. Over the last decades, this has motivated efforts to develop hybrid algorithms that can more efficiently couple continuum and particle representations in a variety of fields, including hypersonic gas dynamics (Schwartzentruber & Boyd, 2006) , high-energy-density physics (Fiuza et al., 2011) , and plasma physics (Bai et al., 2015) . Plasmas -hot ionized gases of charged particles -are a particularly challenging class of complex physics systems, where long-range electromagnetic interactions inevitably drive multi-scale and far-from-equilibrium dynamics. Indeed, plasma research associated with controlled nuclear fu-  I 6 u D Y E 3 O 8 v H J U i r z Y 1 s n K U V 4 = " > A A A B + H i c b V D L S s N A F J 3 U V 6 2 P R l 2 6 C R b B V U l E 0 W X B j b i q Y B / Q h j C Z 3 r R j J 5 M w c y P W 0 i 9 x 4 0 I R t 3 6 K O / / G a Z u F t h 4 Y O J x z L 3 P P C V P B N b r u t 1 V Y W V 1 b 3 y h u l r a 2 d 3 b L 9 t 5 + U y e Z Y t B g i U h U O 6 Q a B J f Q Q I 4 C 2 q k C G o c C W u H w a u q 3 H k B p n s g 7 H K X g x 7 Q v e c Q Z R S M F d r k f d B E e c c z l P T C c B H b F r b o z O M v E y 0 m F 5 K g H 9 l e 3 l 7 A s B o l M U K 0 7 n p u i P 6 Y K O R M w K X U z D S l l Q 9 q H j q G S x q D 9 8 e z w i X N s l J 4 T J c o 8 i c 5 M / b 0 x p r H W o z g 0 k z H F g V 7 0 p u J / X i f D 6 N I 3 m d I M Q b L 5 R 1 E m H E y c a Q t O j y u T V o w M o U x x c 6 v D B l R R h q a r k i n B W 4 y 8 T J q n V e + s e n 5 7 V q n d 5 H U U y S E 5 I i f E I x e k R q 5 J n T Q I I x l 5 J q / k z X q y X q x 3 6 2 M + W r D y n Q P y B 9 b n D 4 H 1 k 6 0 = < / l a t e x i t > g inject < l a t e x i t s h a 1 _ b a s e 6 4 = " b g H I 6 u D Y E 3 O 8 v H J U i r z Y 1 s n K U V 4 = " > A A A B + H i c b V D L S s N A F J 3 U V 6 2 P R l 2 6 C R b B V U l E 0 W X B j b i q Y B / Q h j C Z 3 r R j J 5 M w c y P W 0 i 9 x 4 0 I R t 3 6 K O / / G a Z u F t h 4 Y O J x z L 3 P P C V P B N b r u t 1 V Y W V 1 b 3 y h u l r a 2 d 3 b L 9 t 5 + U y e Z Y t B g i U h U O 6 Q a B J f Q Q I 4 C 2 q k C G o c C W u H w a u q 3 H k B p n s g 7 H K X g x 7 Q v e c Q Z R S M F d r k f d B E e c c z l P T C c B H b F r b o z O M v E y 0 m F 5 K g H 9 l e 3 l 7 A s B o l M U K 0 7 n p u i P 6 Y K O R M w K X U z D S l l Q 9 q H j q G S x q D 9 8 e z w i X N s l J 4 T J c o 8 i c 5 M / b 0 x p r H W o z g 0 k z H F g V 7 0 p u J / X i f D 6 N I 3 m d I M Q b L 5 R 1 E m H E y c a Q t O j y u T V o w M o U x x c 6 v D B l R R h q a r k i n B W 4 y 8 T J q n V e + s e n 5 7 V q n d 5 H U U y S E 5 I i f E I x e k R q 5 J n T Q I I x l 5 J q / k z X q y X q x 3 6 2 M + W r D y n Q P y B 9 b n D 4 H 1 k 6 0 = < / l a t e x i t > g fluid < l a t e x i t s h a 1 _ b a s e 6 4 = " s 1 7 + u s C f I G l / v d 8 n X q 5 d U r L 3 9 z o = " > A A A B 9 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S R S 0 W P B i 3 i q Y D + g j W W z 2 b R L N 5 u w O 1 F L 6 P / w 4 k E R r / 4 X b / 4 b t 2 0 O 2 v p g 4 P H e D D P z / E R w j Y 7 z b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S 8 e p o q x J Y x G r j k 8 0 E 1 y y J n I U r J M o R i J f s L Y / u p r 6 7 Q e m N I / l H Y 4 T 5 k V k I H n I K U E j 3 Q / 6 P W R P m I U i 5 c G k X 6 4 4 V W c G e 5 m 4 O a l A j k a / / N U L Y p p G T C I V R O u u 6 y T o Z U Q h p 4 J N S r 1 U s 4 T Q E R m w r q G S R E x 7 2 e z q i X 1 i l M A O Y 2 V K o j 1 T f 0 9 k J N J 6 H P m m M y I 4 1 I v e V P z P 6 6 Y Y X n o Z l 0 m K T N L 5 o j A V N s b 2 N A I 7 4 I p R F G N D C F X c 3 G r T I V G E o g m q Z E J w F 1 9 e J q 2 z q l u r n t / W K v W b P I 4 i H M E x n I I L F 1 C H a 2 h A E y g o e I Z X e L M e r R f r 3 f q Y t x a s f O Y Q / s D 6 / A F A Q 5 M J < / l a t e x i t > g fluid < l a t e x i t s h a 1 _ b a s e 6 4 = " s 1 7 + u s C f I G l / v d 8 n X q 5 d U r L 3 9 z o = " > A A A B 9 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S R S 0 W P B i 3 i q Y D + g j W W z 2 b R L N 5 u w O 1 F L 6 P / w 4 k E R r / 4 X b / 4 b t 2 0 O 2 v p g 4 P H e D D P z / E R w j Y 7 z b R V W V t f W N 4 q b p a 3 t n d 2 9 8 v 5 B S 8 e p o q x J Y x G r j k 8 0 E 1 y y J n I U r J M o R i J f s L Y / u p r 6 7 Q e m N I / l H Y 4 T 5 k V k I H n I K U E j 3 Q / 6 P W R P m I U i 5 c G k X 6 4 4 V W c G e 5 m 4 O a l A j k a / / N U L Y p p G T C I V R O u u 6 y T o Z U Q h p 4 J N S r 1 U s 4 T Q E R m w r q G S R E x 7 2 e z q i X 1 i l M A O Y 2 V K o j 1 T f 0 9 k J N J 6 H P m m M y I 4 1 I v e V P z P 6 6 Y Y X n o Z l 0 m K T N L 5 o j A V N s b 2 N A I 7 4 I p R F G N D C F X c 3 G r T I V G E o g m q Z E J w F 1 9 e J q 2 z q l u r n t / W K v W b P I 4 i H M E x n I I L F 1 C H a 2 h A E y g o e I Z X e L M e r R f r 3 f q Y t x a s f O Y Q / s D 6 / A F A Q 5 M J < / l a t e x i t > ginject < l a t e x i t s h a 1 _ b a s e 6 4 = " b g H I 6 u D Y E 3 O 8 v H J U i r z Y 1 s n K U V 4 = " > A A A B + H i c b V D L S s N A F J 3 U V 6 2 P R l 2 6 C R b B V U l E 0 W X B j b i q Y B / Q h j C Z 3 r R j J 5 M w c y P W 0 i 9 x 4 0 I R t 3 6 K O / / G a Z u F t h 4 Y O J x z L 3 P P C V P B N b r u t 1 V Y W V 1 b 3 y h u l r a 2 d 3 b L 9 t 5 + U y e Z Y t B g i U h U O 6 Q a B J f Q Q I 4 C 2 q k C G o c C W u H w a u q 3 H k B p n s g 7 H K X g x 7 Q v e c Q Z R S M F d r k f d B E e c c z l P T C c B H b F r b o z O M v E y 0 m F 5 K g H 9 l e 3 l 7 A s B o l M U K 0 7 n p u i P 6 Y K O R M w K X U z D S l l Q 9 q H j q G S x q D 9 8 e z w i X N s l J 4 T J c o 8 i c 5 M / b 0 x p r H W o z g 0 k z H F g V 7 0 p u J / X i f D 6 N I 3 m d I M Q b L 5 R 1 E m H E y c a Q t O j y u T V o w M o U x x c 6 v D B l R R h q a r k i n B W 4 y 8 T J q n V e + s e n 5 7 V q n d 5 H U U y S E 5 I i f E I x e k R q 5 J n T Q I I x l 5 J q / k z X q y X q x 3 6 2 M + W r D y n Q P y B 9 b n D 4 H 1 k 6 0 = < / l a t e x i t > , 1985; Fiuza et al., 2011; Bai et al., 2015; Liu et al., 2022) . In all these cases, small-scale kinetic processes can accelerate a very small group of particles to energies significantly above the mean (thermal) energy, driving the system out of equilibrium. Importantly, this small group (few %) of non-thermal particles can carry away a large fraction (up to 50%) of the system energy and thus impact its global evolution. This motivated the recent development of hybrid representations that use a fluid solver to model the near-equilibrium (thermal) part of the particle distribution and a particle-based kinetic solver to model the evolution of high-energy particles (Kowal et al., 2011; Bai et al., 2015; Guidoni et al., 2016) . However, the coupling of the two representations has been based on over-simplified phenomenological prescriptions that can limit their validity and applicability. = " > A A A B 8 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s x I R Z c F N + K q g n 1 g O 5 R M e q c N z W S G J C O U o X / h x o U i b v 0 b d / 6 N a T s L b T 0 Q O J x z L z n 3 B I n g 2 r j u t 1 N Y W 9 / Y 3 C p u l 3 Z 2 9 / Y P y o d H L R 2 n i m G T x S J W n Y B q F F x i 0 3 A j s J M o p F E g s B 2 M b 2 Z + + w m V 5 r F 8 M J M E / Y g O J Q 8 5 o 8 Z K j 7 2 I m l E Q Z u m 0 X 6 6 4 V X c O s k q 8 n F Q g R 6 N f / u o N Y p Z G K A 0 T V O u u 5 y b G z 6 g y n A m c l n q p x o S y M R 1 i 1 1 J J I 9 R + N k 8 8 J W d W G Z A w V v Z J Q + b q 7 4 2 M R l p P o s B O z h L q Z W 8 m / u d 1 U x N e + x m X S W p Q s s V H Y S q I i c n s f D L g C p k R E 0 s o U 9 x m J W x E F W X G l l S y J X j L J 6 + S 1 k X V q 1 U v 7 2 u V + l 1 e R x F O 4 B T O w Y M r q M M t N K A J D C Q 8 w = " > A A A B 8 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s x I R Z c F N + K q g n 1 g O 5 R M e q c N z W S G J C O U o X / h x o U i b v 0 b d / 6 N a T s L b T 0 Q O J x z L z n 3 B I n g 2 r j u t 1 N Y W 9 / Y 3 C p u l 3 Z 2 9 / Y P y o d H L R 2 n i m G T x S J W n Y B q F F x i 0 3 A j s J M o p F E g s B 2 M b 2 Z + + w m V 5 r F 8 M J M E / Y g O J Q 8 5 o 8 Z K j 7 2 I m l E Q Z u m 0 X 6 6 4 V X c O s k q 8 n F Q g R 6 N f / u o N Y p Z G K A 0 T V O u u 5 y b G z 6 g y n A m c l n q p x o S y M R 1 i 1 1 J J I 9 R + N k 8 8 J W d W G Z A w V v Z J Q + b q 7 4 2 M R l p P o s B O z h L q Z W 8 m / u d 1 U x N e + x m X S W p Q s s V H Y S q I i c n s f D L g C p k R E 0 s o U 9 x m J W x E F W X G l l S y J X j L J 6 + S 1 k X V q 1 U v 7 2 u V + l 1 e R x F O 4 B T O w Y M r q M M t N K A J D C Q 8 w = " > A A A B 8 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s x I R Z c F N + K q g n 1 g O 5 R M e q c N z W S G J C O U o X / h x o U i b v 0 b d / 6 N a T s L b T 0 Q O J x z L z n 3 B I n g 2 r j u t 1 N Y W 9 / Y 3 C p u l 3 Z 2 9 / Y P y o d H L R 2 n i m G T x S J W n Y B q F F x i 0 3 A j s J M o p F E g s B 2 M b 2 Z + + w m V 5 r F 8 M J M E / Y g O J Q 8 5 o 8 Z K j 7 2 I m l E Q Z u m 0 X 6 6 4 V X c O s k q 8 n F Q g R 6 N f / u o N Y p Z G K A 0 T V O u u 5 y b G z 6 g y n A m c l n q p x o S y M R 1 i 1 1 J J I 9 R + N k 8 8 J W d W G Z A w V v Z J Q + b q 7 4 2 M R l p P o s B O z h L q Z W 8 m / u d 1 U x N e + x m X S W p Q s s V H Y S q I i c n s f D L g C p k R E 0 s o U 9 x m J W x E F W X G l l S y J X j L J 6 + S 1 k X V q 1 U v 7 2 u V + l 1 e R x F O 4 B T O w Y M r q M M t N K A J D C Q 8 w y u 8 O d p 5 c d 6 d j 8 V o w c l 3 j u E P n M 8 f / a q R K Q = = < / l a t e x i t > u < l a t e x i t s h a 1 _ b a s e 6 4 = " o 4 u D T 5 G 0 z T H J L Z r w I T D N 6 T p I X j U = " > A A A B 8 X i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s x I R Z c F N + K q g n 1 g O 5 R M e q c N z W S G J C O U o X / h x o U i b v 0 b d / 6 N a T s L b T 0 Q O J x z L z n 3 B I n g 2 r j u t 1 N Y W 9 / Y 3 C p u l 3 Z 2 9 / Y P y o d H L R 2 n i m G T x S J W n Y B q F F x i 0 3 A j s J M o p F E g s B 2 M b 2 Z + + w m V 5 r F 8 M J M E / Y g O J Q 8 5 o 8 Z K j 7 2 I m l E Q Z u m 0 X 6 6 4 V X c O s k q 8 n F Q g R 6 N f / u o N Y p Z G K A 0 T V O u u 5 y b G z 6 g y n A m c l n q p x o S y M R 1 i 1 1 J J I 9 R + N k 8 8 J W d W G Z A w V v Z J Q + b q 7 4 2 M R l p P o s B O z h L q Z W 8 m / u d 1 U x N e + x m X S W p Q s s V H Y S q I i c n s f D L g C p k R E 0 s o U 9 x m J W x E F W X G l l S y J X j L J 6 + S 1 k X V q 1 U v 7 2 u V + l 1 e R x F O 4 B T O w Y M r q M M t N K A J D C Q 8 w In this work, we introduce a method for Learning Hybrid Particle-Continuum (LHPC) models to address the key challenge of efficiently and accurately coupling fluid and kinetic representations of far-from-equilibrium N-body systems. LHPC combines a classical particle-in-cell (PIC) solver to advance the non-thermal particle distribution with a neural network surrogate model that efficiently advances the thermal component using a continuum representation. Most importantly, our key contribution is the use of an additional neural network to learn the self-consistent particle-continuum coupling: the dynamical exchange of mass, momentum, and energy between the particle and continuum representations. This coupling is learned from the data of first-principles PIC simulations, providing an accurate and physics-informed description that addresses the main limitation of previous hybrid methods. While the combination of classical numerical solvers and deep learning has been explored before (Um et al., 2020; Vlachas et al., 2022) , these have been primarily based on uniform representations. The use of machine learning techniques to learn efficient and accurate coupling between continuum and particle representations introduced in this work is a promising and important route for addressing the multi-scale and multi-physics challenge of modeling N-body non-equilibrium systems. We demonstrate our method on a challenging non-linear, far-from-equilibrium N-body system: the interaction of an intense laser with a solid-density plasma and resulting particle acceleration. Present-day high-power lasers reach intensities in excess of 10 20 W/cm 2 , which nearly instantaneously vaporize and ionize solid-state matter upon interaction, resulting in high-energy-density plasma. These interactions give rise to nonlinear and kinetic processes that establish strong elec-



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Figure 1: (a) Schematic of our LHPC architecture. It consists of three components: (1) a neural network g fluid for evolution of thermal sub-population with fluid representation; (2) a solver g * for the evolution of the non-thermal energetic sub-population with particle representation; (3) neural networks that model the injection of particles from fluid state (g inject ). (b) and (c) illustrate the architectures of the components (1) and (3).

